diff --git a/codes/c/chapter_searching/binary_search_edge.c b/codes/c/chapter_searching/binary_search_edge.c
deleted file mode 100644
index 077317ca..00000000
--- a/codes/c/chapter_searching/binary_search_edge.c
+++ /dev/null
@@ -1,59 +0,0 @@
-/**
- * File: binary_search_edge.c
- * Created Time: 2023-05-31
- * Author: Gonglja (glj0@outlook.com)
- */
-
-#include "../utils/common.h"
-
-
-/* 二分查找最左一个元素 */
-int binarySearchLeftEdge(int *nums, int size, int target) {
-    int i = 0, j = size - 1; // 初始化双闭区间 [0, n-1]
-    while (i <= j) {
-        int m = i + (j - i) / 2; // 计算中点索引 m
-        if (nums[m] < target)
-            i = m + 1; // target 在区间 [m+1, j] 中
-        else if (nums[m] > target)
-            j = m - 1; // target 在区间 [i, m-1] 中
-        else
-            j = m - 1; // 首个小于 target 的元素在区间 [i, m-1] 中
-    }
-    if (i == size || nums[i] != target)
-        return -1; // 未找到目标元素，返回 -1
-    return i;
-}
-
-/* 二分查找最右一个元素 */
-int binarySearchRightEdge(int *nums, int size, int target) {
-    int i = 0, j = size - 1; // 初始化双闭区间 [0, n-1]
-    while (i <= j) {
-        int m = i + (j - i) / 2; // 计算中点索引 m
-        if (nums[m] < target)
-            i = m + 1; // target 在区间 [m+1, j] 中
-        else if (nums[m] > target)
-            j = m - 1; // target 在区间 [i, m-1] 中
-        else
-            i = m + 1; // 首个大于 target 的元素在区间 [m+1, j] 中
-    }
-    if (j < 0 || nums[j] != target)
-        return -1; // 未找到目标元素，返回 -1
-    return j;
-}
-
-/* Driver Code */
-int main() {
-    int target = 6;
-    int nums[] = {1, 3, 6, 6, 6, 6, 6, 10, 12, 15};
-    int size = sizeof(nums) / sizeof(nums[0]);
-
-    // 二分查找最左一个元素
-    int indexLeft = binarySearchLeftEdge(nums, size, target);
-    printf("数组中最左一个元素 6 的索引 = %d\n", indexLeft);
-
-    // 二分查找最右一个元素
-    int indexRight = binarySearchRightEdge(nums, size, target);
-    printf("数组中最右一个元素 6 的索引 = %d\n", indexRight);
-
-    return 0;
-}
\ No newline at end of file
diff --git a/codes/cpp/chapter_searching/CMakeLists.txt b/codes/cpp/chapter_searching/CMakeLists.txt
index d65ee314..60a223d8 100644
--- a/codes/cpp/chapter_searching/CMakeLists.txt
+++ b/codes/cpp/chapter_searching/CMakeLists.txt
@@ -1,3 +1,4 @@
 add_executable(binary_search binary_search.cpp)
+add_executable(binary_search_insertion binary_search_insertion.cpp)
 add_executable(binary_search_edge binary_search_edge.cpp)
 add_executable(two_sum two_sum.cpp)
diff --git a/codes/cpp/chapter_searching/binary_search_edge.cpp b/codes/cpp/chapter_searching/binary_search_edge.cpp
index 332fe6c1..494e380d 100644
--- a/codes/cpp/chapter_searching/binary_search_edge.cpp
+++ b/codes/cpp/chapter_searching/binary_search_edge.cpp
@@ -1,57 +1,66 @@
 /**
  * File: binary_search_edge.cpp
- * Created Time: 2023-05-21
+ * Created Time: 2023-08-04
  * Author: Krahets (krahets@163.com)
  */
 
 #include "../utils/common.hpp"
 
-/* 二分查找最左一个元素 */
-int binarySearchLeftEdge(vector<int> &nums, int target) {
+/* 二分查找插入点（存在重复元素） */
+int binarySearchInsertion(const vector<int> &nums, int target) {
     int i = 0, j = nums.size() - 1; // 初始化双闭区间 [0, n-1]
     while (i <= j) {
         int m = i + (j - i) / 2; // 计算中点索引 m
-        if (nums[m] < target)
+        if (nums[m] < target) {
             i = m + 1; // target 在区间 [m+1, j] 中
-        else if (nums[m] > target)
-            j = m - 1; // target 在区间 [i, m-1] 中
-        else
+        } else {
             j = m - 1; // 首个小于 target 的元素在区间 [i, m-1] 中
+        }
     }
-    if (i == nums.size() || nums[i] != target)
-        return -1; // 未找到目标元素，返回 -1
+    // 返回插入点 i
     return i;
 }
 
-/* 二分查找最右一个元素 */
-int binarySearchRightEdge(vector<int> &nums, int target) {
-    int i = 0, j = nums.size() - 1; // 初始化双闭区间 [0, n-1]
-    while (i <= j) {
-        int m = i + (j - i) / 2; // 计算中点索引 m
-        if (nums[m] < target)
-            i = m + 1; // target 在区间 [m+1, j] 中
-        else if (nums[m] > target)
-            j = m - 1; // target 在区间 [i, m-1] 中
-        else
-            i = m + 1; // 首个大于 target 的元素在区间 [m+1, j] 中
+/* 二分查找最左一个 target */
+int binarySearchLeftEdge(vector<int> &nums, int target) {
+    // 等价于查找 target 的插入点
+    int i = binarySearchInsertion(nums, target);
+    // 未找到 target ，返回 -1
+    if (i == nums.size() || nums[i] != target) {
+        return -1;
     }
-    if (j < 0 || nums[j] != target)
-        return -1; // 未找到目标元素，返回 -1
+    // 找到 target ，返回索引 i
+    return i;
+}
+
+/* 二分查找最右一个 target */
+int binarySearchRightEdge(vector<int> &nums, int target) {
+    // 转化为查找最左一个 target + 1
+    int i = binarySearchInsertion(nums, target + 1);
+    // j 指向最右一个 target ，i 指向首个大于 target 的元素
+    int j = i - 1;
+    // 未找到 target ，返回 -1
+    if (j == -1 || nums[j] != target) {
+        return -1;
+    }
+    // 找到 target ，返回索引 j
     return j;
 }
 
 /* Driver Code */
 int main() {
-    int target = 6;
+    // 包含重复元素的数组
     vector<int> nums = {1, 3, 6, 6, 6, 6, 6, 10, 12, 15};
+    cout << "\n数组 nums = ";
+    printVector(nums);
 
-    // 二分查找最左一个元素
-    int indexLeft = binarySearchLeftEdge(nums, target);
-    cout << "数组中最左一个元素 6 的索引 = " << indexLeft << endl;
-
-    // 二分查找最右一个元素
-    int indexRight = binarySearchRightEdge(nums, target);
-    cout << "数组中最右一个元素 6 的索引 = " << indexRight << endl;
+    // 二分查找左边界和右边界
+    for (int target : {6, 7}) {
+        int index = binarySearchLeftEdge(nums, target);
+        cout << "最左一个元素 " << target << " 的索引为 " << index << endl;
+        index = binarySearchRightEdge(nums, target);
+        cout << "最右一个元素 " << target << " 的索引为 " << index << endl;
+    }
 
     return 0;
 }
diff --git a/codes/cpp/chapter_searching/binary_search_insertion.cpp b/codes/cpp/chapter_searching/binary_search_insertion.cpp
new file mode 100644
index 00000000..a5e1a178
--- /dev/null
+++ b/codes/cpp/chapter_searching/binary_search_insertion.cpp
@@ -0,0 +1,66 @@
+/**
+ * File: binary_search_edge.cpp
+ * Created Time: 2023-08-04
+ * Author: Krahets (krahets@163.com)
+ */
+
+#include "../utils/common.hpp"
+
+/* 二分查找插入点（无重复元素） */
+int binarySearchInsertionSimple(vector<int> &nums, int target) {
+    int i = 0, j = nums.size() - 1; // 初始化双闭区间 [0, n-1]
+    while (i <= j) {
+        int m = i + (j - i) / 2; // 计算中点索引 m
+        if (nums[m] < target) {
+            i = m + 1; // target 在区间 [m+1, j] 中
+        } else if (nums[m] > target) {
+            j = m - 1; // target 在区间 [i, m-1] 中
+        } else {
+            return m; // 找到 target ，返回插入点 m
+        }
+    }
+    // 未找到 target ，返回插入点 i
+    return i;
+}
+
+/* 二分查找插入点（存在重复元素） */
+int binarySearchInsertion(vector<int> &nums, int target) {
+    int i = 0, j = nums.size() - 1; // 初始化双闭区间 [0, n-1]
+    while (i <= j) {
+        int m = i + (j - i) / 2; // 计算中点索引 m
+        if (nums[m] < target) {
+            i = m + 1; // target 在区间 [m+1, j] 中
+        } else if (nums[m] > target) {
+            j = m - 1; // target 在区间 [i, m-1] 中
+        } else {
+            j = m - 1; // 首个小于 target 的元素在区间 [i, m-1] 中
+        }
+    }
+    // 返回插入点 i
+    return i;
+}
+
+/* Driver Code */
+int main() {
+    // 无重复元素的数组
+    vector<int> nums = {1, 3, 6, 8, 12, 15, 23, 26, 31, 35};
+    cout << "\n数组 nums = ";
+    printVector(nums);
+    // 二分查找插入点
+    for (int target : {6, 9}) {
+        int index = binarySearchInsertionSimple(nums, target);
+        cout << "元素 " << target << " 的插入点的索引为 " << index << endl;
+    }
+
+    // 包含重复元素的数组
+    nums = {1, 3, 6, 6, 6, 6, 6, 10, 12, 15};
+    cout << "\n数组 nums = ";
+    printVector(nums);
+    // 二分查找插入点
+    for (int target : {2, 6, 20}) {
+        int index = binarySearchInsertion(nums, target);
+        cout << "元素 " << target << " 的插入点的索引为 " << index << endl;
+    }
+
+    return 0;
+}
diff --git a/codes/csharp/chapter_searching/binary_search_edge.cs b/codes/csharp/chapter_searching/binary_search_edge.cs
deleted file mode 100644
index 081cc6a1..00000000
--- a/codes/csharp/chapter_searching/binary_search_edge.cs
+++ /dev/null
@@ -1,57 +0,0 @@
-/**
-* File: binary_search_edge.cs
-* Created Time: 2023-06-01
-* Author: hpstory (hpstory1024@163.com)
-*/
-
-namespace hello_algo.chapter_searching; 
-
-public class binary_search_edge {
-    /* 二分查找最左一个元素 */
-    public static int binarySearchLeftEdge(int[] nums, int target) {
-        int i = 0, j = nums.Length - 1; // 初始化双闭区间 [0, n-1]
-        while (i <= j) {
-            int m = i + (j - i) / 2; // 计算中点索引 m
-            if (nums[m] < target)
-                i = m + 1; // target 在区间 [m+1, j] 中
-            else if (nums[m] > target)
-                j = m - 1; // target 在区间 [i, m-1] 中
-            else
-                j = m - 1; // 首个小于 target 的元素在区间 [i, m-1] 中
-        }
-        if (i == nums.Length || nums[i] != target)
-            return -1; // 未找到目标元素，返回 -1
-        return i;
-    }
-
-    /* 二分查找最右一个元素 */
-    public static int binarySearchRightEdge(int[] nums, int target) {
-        int i = 0, j = nums.Length - 1; // 初始化双闭区间 [0, n-1]
-        while (i <= j) {
-            int m = i + (j - i) / 2; // 计算中点索引 m
-            if (nums[m] < target)
-                i = m + 1; // target 在区间 [m+1, j] 中
-            else if (nums[m] > target)
-                j = m - 1; // target 在区间 [i, m-1] 中
-            else
-                i = m + 1; // 首个大于 target 的元素在区间 [m+1, j] 中
-        }
-        if (j < 0 || nums[j] != target)
-            return -1; // 未找到目标元素，返回 -1
-        return j;
-    }
-
-    [Test]
-    public void Test() {
-        int target = 6;
-        int[] nums = { 1, 3, 6, 6, 6, 6, 6, 10, 12, 15 };
-
-        // 二分查找最左一个元素
-        int indexLeft = binarySearchLeftEdge(nums, target);
-        Console.WriteLine("数组中最左一个元素 6 的索引 = " + indexLeft);
-
-        // 二分查找最右一个元素
-        int indexRight = binarySearchRightEdge(nums, target);
-        Console.WriteLine("数组中最右一个元素 6 的索引 = " + indexRight);
-    }
-}
diff --git a/codes/dart/chapter_searching/binary_search_edge.dart b/codes/dart/chapter_searching/binary_search_edge.dart
deleted file mode 100644
index 4b248131..00000000
--- a/codes/dart/chapter_searching/binary_search_edge.dart
+++ /dev/null
@@ -1,50 +0,0 @@
-/**
- * File: binary_search_edge.dart
- * Created Time: 2023-06-01
- * Author: liuyuxin (gvenusleo@gmail.com)
- */
-
-/* 二分查找最左一个元素 */
-int binarySearchLeftEdge(List<int> nums, int target) {
-  int i = 0, j = nums.length - 1; // 初始化双闭区间 [0, n-1]
-  while (i <= j) {
-    int m = i + (j - i) ~/ 2; // 计算中间索引 m
-    if (nums[m] < target)
-      i = m + 1; // target 在区间 [m+1, j] 中
-    else if (nums[m] > target)
-      j = m - 1; // target 在区间 [i, m-1] 中
-    else
-      j = m - 1; // 首个小于 target 的元素在区间 [i, m-1] 中
-  }
-  if (i == nums.length || nums[i] != target) return -1; // 未找到目标元素，返回 -1
-  return i;
-}
-
-/* 二分查找最右一个元素 */
-int binarySearchRightEdge(List<int> nums, int target) {
-  int i = 0, j = nums.length - 1; // 初始化双闭区间 [0, n-1]
-  while (i <= j) {
-    int m = i + (j - i) ~/ 2; // 计算中间索引 m
-    if (nums[m] < target)
-      i = m + 1; // target 在区间 [m+1, j] 中
-    else if (nums[m] > target)
-      j = m - 1; // target 在区间 [i, m-1] 中
-    else
-      i = m + 1; // 首个大于 target 的元素在区间 [m+1, j] 中
-  }
-  if (j < 0 || nums[j] != target) return -1; // 未找到目标元素，返回 -1
-  return j;
-}
-
-/* Driver Code */
-void main() {
-  int target = 6;
-  List<int> nums = [1, 3, 6, 6, 6, 6, 6, 10, 12, 15];
-
-  // 二分查找最左一个元素
-  int indexLeft = binarySearchLeftEdge(nums, target);
-  print("数组中最左一个元素 6 的索引 = $indexLeft");
-  // 二分查找最右一个元素
-  int indexRight = binarySearchRightEdge(nums, target);
-  print("数组中最右一个元素 6 的索引 = $indexRight");
-}
diff --git a/codes/go/chapter_searching/binary_search_edge.go b/codes/go/chapter_searching/binary_search_edge.go
deleted file mode 100644
index 8a27a626..00000000
--- a/codes/go/chapter_searching/binary_search_edge.go
+++ /dev/null
@@ -1,55 +0,0 @@
-// File: binary_search_edge.go
-// Created Time: 2023-05-29
-// Author: Reanon (793584285@qq.com)
-
-package chapter_searching
-
-/* 二分查找最左一个元素 */
-func binarySearchLeftEdge(nums []int, target int) int {
-	// 初始化双闭区间 [0, n-1]
-	i, j := 0, len(nums)-1
-	for i <= j {
-		// 计算中点索引 m
-		m := i + (j-i)/2
-		if nums[m] < target {
-			// target 在区间 [m+1, j] 中
-			i = m + 1
-		} else if nums[m] > target {
-			// target 在区间 [i, m-1] 中
-			j = m - 1
-		} else {
-			// 首个小于 target 的元素在区间 [i, m-1] 中
-			j = m - 1
-		}
-	}
-	if i == len(nums) || nums[i] != target {
-		// 未找到目标元素，返回 -1
-		return -1
-	}
-	return i
-}
-
-/* 二分查找最右一个元素 */
-func binarySearchRightEdge(nums []int, target int) int {
-	// 初始化双闭区间 [0, n-1]
-	i, j := 0, len(nums)-1
-	for i <= j {
-		// 计算中点索引 m
-		m := i + (j-i)/2
-		if nums[m] < target {
-			// target 在区间 [m+1, j] 中
-			i = m + 1
-		} else if nums[m] > target {
-			// target 在区间 [i, m-1] 中
-			j = m - 1
-		} else {
-			// 首个大于 target 的元素在区间 [m+1, j] 中
-			i = m + 1
-		}
-	}
-	if j < 0 || nums[j] != target {
-		// 未找到目标元素，返回 -1
-		return -1
-	}
-	return j
-}
diff --git a/codes/go/chapter_searching/binary_search_test.go b/codes/go/chapter_searching/binary_search_test.go
index 74d10505..0b6084d6 100644
--- a/codes/go/chapter_searching/binary_search_test.go
+++ b/codes/go/chapter_searching/binary_search_test.go
@@ -22,15 +22,3 @@ func TestBinarySearch(t *testing.T) {
 		t.Errorf("目标元素 6 的索引 = %d, 应该为 %d", actual, expected)
 	}
 }
-
-func TestBinarySearchEdge(t *testing.T) {
-	target := 6
-	nums := []int{1, 3, 6, 6, 6, 6, 6, 10, 12, 15}
-	// 二分查找最左一个元素
-	indexLeft := binarySearchLeftEdge(nums, target)
-	fmt.Println("数组中最左一个元素 6 的索引 = ", indexLeft)
-
-	// 二分查找最右一个元素
-	indexRight := binarySearchRightEdge(nums, target)
-	fmt.Println("数组中最右一个元素 6 的索引 = ", indexRight)
-}
diff --git a/codes/java/chapter_searching/binary_search_edge.java b/codes/java/chapter_searching/binary_search_edge.java
index 847a41e8..c23b655e 100644
--- a/codes/java/chapter_searching/binary_search_edge.java
+++ b/codes/java/chapter_searching/binary_search_edge.java
@@ -1,56 +1,49 @@
 /**
  * File: binary_search_edge.java
- * Created Time: 2023-05-21
+ * Created Time: 2023-08-04
  * Author: Krahets (krahets@163.com)
  */
 
 package chapter_searching;
 
 public class binary_search_edge {
-    /* 二分查找最左一个元素 */
+    /* 二分查找最左一个 target */
     static int binarySearchLeftEdge(int[] nums, int target) {
-        int i = 0, j = nums.length - 1; // 初始化双闭区间 [0, n-1]
-        while (i <= j) {
-            int m = i + (j - i) / 2; // 计算中点索引 m
-            if (nums[m] < target)
-                i = m + 1; // target 在区间 [m+1, j] 中
-            else if (nums[m] > target)
-                j = m - 1; // target 在区间 [i, m-1] 中
-            else
-                j = m - 1; // 首个小于 target 的元素在区间 [i, m-1] 中
+        // 等价于查找 target 的插入点
+        int i = binary_search_insertion.binarySearchInsertion(nums, target);
+        // 未找到 target ，返回 -1
+        if (i == nums.length || nums[i] != target) {
+            return -1;
         }
-        if (i == nums.length || nums[i] != target)
-            return -1; // 未找到目标元素，返回 -1
+        // 找到 target ，返回索引 i
         return i;
     }
 
-    /* 二分查找最右一个元素 */
+    /* 二分查找最右一个 target */
     static int binarySearchRightEdge(int[] nums, int target) {
-        int i = 0, j = nums.length - 1; // 初始化双闭区间 [0, n-1]
-        while (i <= j) {
-            int m = i + (j - i) / 2; // 计算中点索引 m
-            if (nums[m] < target)
-                i = m + 1; // target 在区间 [m+1, j] 中
-            else if (nums[m] > target)
-                j = m - 1; // target 在区间 [i, m-1] 中
-            else
-                i = m + 1; // 首个大于 target 的元素在区间 [m+1, j] 中
+        // 转化为查找最左一个 target + 1
+        int i = binary_search_insertion.binarySearchInsertion(nums, target + 1);
+        // j 指向最右一个 target ，i 指向首个大于 target 的元素
+        int j = i - 1;
+        // 未找到 target ，返回 -1
+        if (j == -1 || nums[j] != target) {
+            return -1;
         }
-        if (j < 0 || nums[j] != target)
-            return -1; // 未找到目标元素，返回 -1
+        // 找到 target ，返回索引 j
         return j;
     }
 
     public static void main(String[] args) {
-        int target = 6;
+        // 包含重复元素的数组
         int[] nums = { 1, 3, 6, 6, 6, 6, 6, 10, 12, 15 };
+        System.out.println("\n数组 nums = " + java.util.Arrays.toString(nums));
 
-        // 二分查找最左一个元素
-        int indexLeft = binarySearchLeftEdge(nums, target);
-        System.out.println("数组中最左一个元素 6 的索引 = " + indexLeft);
-
-        // 二分查找最右一个元素
-        int indexRight = binarySearchRightEdge(nums, target);
-        System.out.println("数组中最右一个元素 6 的索引 = " + indexRight);
+        // 二分查找左边界和右边界
+        for (int target : new int[] { 6, 7 }) {
+            int index = binarySearchLeftEdge(nums, target);
+            System.out.println("最左一个元素 " + target + " 的索引为 " + index);
+            index = binarySearchRightEdge(nums, target);
+            System.out.println("最右一个元素 " + target + " 的索引为 " + index);
+        }
     }
 }
diff --git a/codes/java/chapter_searching/binary_search_insertion.java b/codes/java/chapter_searching/binary_search_insertion.java
new file mode 100644
index 00000000..cacf553a
--- /dev/null
+++ b/codes/java/chapter_searching/binary_search_insertion.java
@@ -0,0 +1,63 @@
+/**
+ * File: binary_search_edge.java
+ * Created Time: 2023-08-04
+ * Author: Krahets (krahets@163.com)
+ */
+
+package chapter_searching;
+
+class binary_search_insertion {
+    /* 二分查找插入点（无重复元素） */
+    static int binarySearchInsertionSimple(int[] nums, int target) {
+        int i = 0, j = nums.length - 1; // 初始化双闭区间 [0, n-1]
+        while (i <= j) {
+            int m = i + (j - i) / 2; // 计算中点索引 m
+            if (nums[m] < target) {
+                i = m + 1; // target 在区间 [m+1, j] 中
+            } else if (nums[m] > target) {
+                j = m - 1; // target 在区间 [i, m-1] 中
+            } else {
+                return m; // 找到 target ，返回插入点 m
+            }
+        }
+        // 未找到 target ，返回插入点 i
+        return i;
+    }
+
+    /* 二分查找插入点（存在重复元素） */
+    static int binarySearchInsertion(int[] nums, int target) {
+        int i = 0, j = nums.length - 1; // 初始化双闭区间 [0, n-1]
+        while (i <= j) {
+            int m = i + (j - i) / 2; // 计算中点索引 m
+            if (nums[m] < target) {
+                i = m + 1; // target 在区间 [m+1, j] 中
+            } else if (nums[m] > target) {
+                j = m - 1; // target 在区间 [i, m-1] 中
+            } else {
+                j = m - 1; // 首个小于 target 的元素在区间 [i, m-1] 中
+            }
+        }
+        // 返回插入点 i
+        return i;
+    }
+
+    public static void main(String[] args) {
+        // 无重复元素的数组
+        int[] nums = { 1, 3, 6, 8, 12, 15, 23, 26, 31, 35 };
+        System.out.println("\n数组 nums = " + java.util.Arrays.toString(nums));
+        // 二分查找插入点
+        for (int target : new int[] { 6, 9 }) {
+            int index = binarySearchInsertionSimple(nums, target);
+            System.out.println("元素 " + target + " 的插入点的索引为 " + index);
+        }
+
+        // 包含重复元素的数组
+        nums = new int[] { 1, 3, 6, 6, 6, 6, 6, 10, 12, 15 };
+        System.out.println("\n数组 nums = " + java.util.Arrays.toString(nums));
+        // 二分查找插入点
+        for (int target : new int[] { 2, 6, 20 }) {
+            int index = binarySearchInsertion(nums, target);
+            System.out.println("元素 " + target + " 的插入点的索引为 " + index);
+        }
+    }
+}
diff --git a/codes/javascript/chapter_searching/binary_search_edge.js b/codes/javascript/chapter_searching/binary_search_edge.js
deleted file mode 100644
index 4c7aadb9..00000000
--- a/codes/javascript/chapter_searching/binary_search_edge.js
+++ /dev/null
@@ -1,57 +0,0 @@
-/**
- * File: binary_search_edge.js
- * Created Time: 2023-06-04
- * Author: Justin (xiefahit@gmail.com)
- */
-
-/* 二分查找最左一个元素 */
-function binarySearchLeftEdge(nums, target) {
-    let i = 0,
-        j = nums.length - 1; // 初始化双闭区间 [0, n-1]
-    while (i <= j) {
-        let m = Math.floor((i + j) / 2); // 计算中点索引 m
-        if (nums[m] < target) {
-            i = m + 1; // target 在区间 [m+1, j] 中
-        } else if (nums[m] > target) {
-            j = m - 1; // target 在区间 [i, m-1] 中
-        } else {
-            j = m - 1; // 首个小于 target 的元素在区间 [i, m-1] 中
-        }
-    }
-    if (i === nums.length || nums[i] != target) {
-        return -1; // 未找到目标元素，返回 -1
-    }
-    return i;
-}
-
-/* 二分查找最右一个元素 */
-function binarySearchRightEdge(nums, target) {
-    let i = 0,
-        j = nums.length - 1; // 初始化双闭区间 [0, n-1]
-    while (i <= j) {
-        let m = Math.floor((i + j) / 2); // 计算中点索引 m
-        if (nums[m] < target) {
-            i = m + 1; // target 在区间 [m+1, j] 中
-        } else if (nums[m] > target) {
-            j = m - 1; // target 在区间 [i, m-1] 中
-        } else {
-            i = m + 1; // 首个大于 target 的元素在区间 [m+1, j] 中
-        }
-    }
-    if (j < 0 || nums[j] != target) {
-        return -1; // 未找到目标元素，返回 -1
-    }
-    return j;
-}
-
-/* Driver Code */
-let target = 6;
-const nums = [1, 3, 6, 6, 6, 6, 6, 10, 12, 15];
-
-// 二分查找最左一个元素
-let index_left = binarySearchLeftEdge(nums, target);
-console.log('数组中最左一个元素 6 的索引 = ', index_left);
-
-// 二分查找最右一个元素
-let index_right = binarySearchRightEdge(nums, target);
-console.log('数组中最右一个元素 6 的索引 = ', index_right);
diff --git a/codes/python/chapter_searching/binary_search_edge.py b/codes/python/chapter_searching/binary_search_edge.py
index 83e6f1c9..55ae9b8a 100644
--- a/codes/python/chapter_searching/binary_search_edge.py
+++ b/codes/python/chapter_searching/binary_search_edge.py
@@ -1,51 +1,48 @@
 """
 File: binary_search_edge.py
-Created Time: 2023-05-18
+Created Time: 2023-08-04
 Author: Krahets (krahets@163.com)
 """
 
+import sys, os.path as osp
+
+sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
+from binary_search_insertion import binary_search_insertion
+
 
 def binary_search_left_edge(nums: list[int], target: int) -> int:
-    """二分查找最左一个元素"""
-    i, j = 0, len(nums) - 1  # 初始化双闭区间 [0, n-1]
-    while i <= j:
-        m = (i + j) // 2  # 计算中点索引 m
-        if nums[m] < target:
-            i = m + 1  # target 在区间 [m+1, j] 中
-        elif nums[m] > target:
-            j = m - 1  # target 在区间 [i, m-1] 中
-        else:
-            j = m - 1  # 首个小于 target 的元素在区间 [i, m-1] 中
+    """二分查找最左一个 target"""
+    # 等价于查找 target 的插入点
+    i = binary_search_insertion(nums, target)
+    # 未找到 target ，返回 -1
     if i == len(nums) or nums[i] != target:
-        return -1  # 未找到目标元素，返回 -1
+        return -1
+    # 找到 target ，返回索引 i
     return i
 
 
 def binary_search_right_edge(nums: list[int], target: int) -> int:
-    """二分查找最右一个元素"""
-    i, j = 0, len(nums) - 1  # 初始化双闭区间 [0, n-1]
-    while i <= j:
-        m = (i + j) // 2  # 计算中点索引 m
-        if nums[m] < target:
-            i = m + 1  # target 在区间 [m+1, j] 中
-        elif nums[m] > target:
-            j = m - 1  # target 在区间 [i, m-1] 中
-        else:
-            i = m + 1  # 首个大于 target 的元素在区间 [m+1, j] 中
-    if j < 0 or nums[j] != target:
-        return -1  # 未找到目标元素，返回 -1
+    """二分查找最右一个 target"""
+    # 转化为查找最左一个 target + 1
+    i = binary_search_insertion(nums, target + 1)
+    # j 指向最右一个 target ，i 指向首个大于 target 的元素
+    j = i - 1
+    # 未找到 target ，返回 -1
+    if j == -1 or nums[j] != target:
+        return -1
+    # 找到 target ，返回索引 j
     return j
 
 
 """Driver Code"""
 if __name__ == "__main__":
-    target = 6
+    # 包含重复元素的数组
     nums = [1, 3, 6, 6, 6, 6, 6, 10, 12, 15]
+    print(f"\n数组 nums = {nums}")
 
-    # 二分查找最左一个元素
-    index_left = binary_search_left_edge(nums, target)
-    print("数组中最左一个元素 6 的索引 = ", index_left)
-
-    # 二分查找最右一个元素
-    index_right = binary_search_right_edge(nums, target)
-    print("数组中最右一个元素 6 的索引 = ", index_right)
+    # 二分查找左边界和右边界
+    for target in [6, 7]:
+        index = binary_search_left_edge(nums, target)
+        print(f"最左一个元素 {target} 的索引为 {index}")
+        index = binary_search_right_edge(nums, target)
+        print(f"最右一个元素 {target} 的索引为 {index}")
diff --git a/codes/python/chapter_searching/binary_search_insertion.py b/codes/python/chapter_searching/binary_search_insertion.py
new file mode 100644
index 00000000..3d0ef4d8
--- /dev/null
+++ b/codes/python/chapter_searching/binary_search_insertion.py
@@ -0,0 +1,54 @@
+"""
+File: binary_search_insertion.py
+Created Time: 2023-08-04
+Author: Krahets (krahets@163.com)
+"""
+
+
+def binary_search_insertion_simple(nums: list[int], target: int) -> int:
+    """二分查找插入点（无重复元素）"""
+    i, j = 0, len(nums) - 1  # 初始化双闭区间 [0, n-1]
+    while i <= j:
+        m = (i + j) // 2  # 计算中点索引 m
+        if nums[m] < target:
+            i = m + 1  # target 在区间 [m+1, j] 中
+        elif nums[m] > target:
+            j = m - 1  # target 在区间 [i, m-1] 中
+        else:
+            return m  # 找到 target ，返回插入点 m
+    # 未找到 target ，返回插入点 i
+    return i
+
+
+def binary_search_insertion(nums: list[int], target: int) -> int:
+    """二分查找插入点（存在重复元素）"""
+    i, j = 0, len(nums) - 1  # 初始化双闭区间 [0, n-1]
+    while i <= j:
+        m = (i + j) // 2  # 计算中点索引 m
+        if nums[m] < target:
+            i = m + 1  # target 在区间 [m+1, j] 中
+        elif nums[m] > target:
+            j = m - 1  # target 在区间 [i, m-1] 中
+        else:
+            j = m - 1  # 首个小于 target 的元素在区间 [i, m-1] 中
+    # 返回插入点 i
+    return i
+
+
+"""Driver Code"""
+if __name__ == "__main__":
+    # 无重复元素的数组
+    nums = [1, 3, 6, 8, 12, 15, 23, 26, 31, 35]
+    print(f"\n数组 nums = {nums}")
+    # 二分查找插入点
+    for target in [6, 9]:
+        index = binary_search_insertion_simple(nums, target)
+        print(f"元素 {target} 的插入点的索引为 {index}")
+
+    # 包含重复元素的数组
+    nums = [1, 3, 6, 6, 6, 6, 6, 10, 12, 15]
+    print(f"\n数组 nums = {nums}")
+    # 二分查找插入点
+    for target in [2, 6, 20]:
+        index = binary_search_insertion(nums, target)
+        print(f"元素 {target} 的插入点的索引为 {index}")
diff --git a/codes/rust/chapter_searching/binary_search_edge.rs b/codes/rust/chapter_searching/binary_search_edge.rs
deleted file mode 100644
index 3d6048bc..00000000
--- a/codes/rust/chapter_searching/binary_search_edge.rs
+++ /dev/null
@@ -1,59 +0,0 @@
-/*
- * File: binary_search_edge.rs
- * Created Time: 2023-05-31
- * Author: WSL0809 (wslzzy@outlook.com)
- */
-
-/* 二分查找最左一个元素 */
-fn binary_search_left_edge(nums: &[i32], target: i32) -> i32 {
-    let mut i = 0;
-    let mut j = nums.len() as i32 - 1; // 初始化双闭区间 [0, n-1]
-    while i <= j {
-        let m = i + (j - i) / 2; // 计算中点索引 m
-        if nums[m as usize] < target {
-            i = m + 1; // target 在区间 [m+1, j] 中
-        } else if nums[m as usize] > target {
-            j = m - 1; // target 在区间 [i, m-1] 中
-        } else {
-            j = m - 1; // 首个小于 target 的元素在区间 [i, m-1] 中
-        }
-    }
-    if i == nums.len() as i32 || nums[i as usize] != target {
-        return -1; // 未找到目标元素，返回 -1
-    }
-    i
-}
-
-/* 二分查找最右一个元素 */
-fn binary_search_right_edge(nums: &[i32], target: i32) -> i32 {
-    let mut i = 0;
-    let mut j = nums.len() as i32 - 1; // 初始化双闭区间 [0, n-1]
-    while i <= j {
-        let m = i + (j - i) / 2; // 计算中点索引 m
-        if nums[m as usize] < target {
-            i = m + 1; // target 在区间 [m+1, j] 中
-        } else if nums[m as usize] > target {
-            j = m - 1; // target 在区间 [i, m-1] 中
-        } else {
-            i = m + 1; // 首个大于 target 的元素在区间 [m+1, j] 中
-        }
-    }
-    if j < 0 || nums[j as usize] != target {
-        return -1; // 未找到目标元素，返回 -1
-    }
-    j
-}
-
-/* Driver Code */
-pub fn main() {
-    let target = 6;
-    let nums = [1, 3, 6, 6, 6, 6, 6, 10, 12, 15];
-
-    // 二分查找最左一个元素
-    let index_left = binary_search_left_edge(&nums, target);
-    println!("数组中最左一个元素 6 的索引 = {}", index_left);
-
-    // 二分查找最右一个元素
-    let index_right = binary_search_right_edge(&nums, target);
-    println!("数组中最右一个元素 6 的索引 = {}", index_right);
-}
\ No newline at end of file
diff --git a/codes/swift/chapter_searching/binary_search_edge.swift b/codes/swift/chapter_searching/binary_search_edge.swift
deleted file mode 100644
index 504f84f4..00000000
--- a/codes/swift/chapter_searching/binary_search_edge.swift
+++ /dev/null
@@ -1,64 +0,0 @@
-/**
- * File: binary_search_edge.swift
- * Created Time: 2023-05-28
- * Author: nuomi1 (nuomi1@qq.com)
- */
-
-/* 二分查找最左一个元素 */
-func binarySearchLeftEdge(nums: [Int], target: Int) -> Int {
-    // 初始化双闭区间 [0, n-1]
-    var i = 0
-    var j = nums.count - 1
-    while i <= j {
-        let m = i + (j - 1) / 2 // 计算中点索引 m
-        if nums[m] < target {
-            i = m + 1 // target 在区间 [m+1, j] 中
-        } else if nums[m] > target {
-            j = m - 1 // target 在区间 [i, m-1] 中
-        } else {
-            j = m - 1 // 首个小于 target 的元素在区间 [i, m-1] 中
-        }
-    }
-    if i == nums.count || nums[i] != target {
-        return -1 // 未找到目标元素，返回 -1
-    }
-    return i
-}
-
-/* 二分查找最右一个元素 */
-func binarySearchRightEdge(nums: [Int], target: Int) -> Int {
-    // 初始化双闭区间 [0, n-1]
-    var i = 0
-    var j = nums.count - 1
-    while i <= j {
-        let m = i + (j - i) / 2 // 计算中点索引 m
-        if nums[m] < target {
-            i = m + 1 // target 在区间 [m+1, j] 中
-        } else if nums[m] > target {
-            j = m - 1 // target 在区间 [i, m-1] 中
-        } else {
-            i = m + 1 // 首个大于 target 的元素在区间 [m+1, j] 中
-        }
-    }
-    if j < 0 || nums[j] != target {
-        return -1 // 未找到目标元素，返回 -1
-    }
-    return j
-}
-
-@main
-enum BinarySearchEdge {
-    /* Driver Code */
-    static func main() {
-        let target = 6
-        let nums = [1, 3, 6, 6, 6, 6, 6, 10, 12, 15]
-
-        // 二分查找最左一个元素
-        let indexLeft = binarySearchLeftEdge(nums: nums, target: target)
-        print("数组中最左一个元素 6 的索引 = \(indexLeft)")
-
-        // 二分查找最右一个元素
-        let indexRight = binarySearchRightEdge(nums: nums, target: target)
-        print("数组中最右一个元素 6 的索引 = \(indexRight)")
-    }
-}
diff --git a/codes/typescript/chapter_searching/binary_search_edge.ts b/codes/typescript/chapter_searching/binary_search_edge.ts
deleted file mode 100644
index a47bee09..00000000
--- a/codes/typescript/chapter_searching/binary_search_edge.ts
+++ /dev/null
@@ -1,55 +0,0 @@
-/**
- * File: binary_search_edge.ts
- * Created Time: 2023-06-04
- * Author: Justin (xiefahit@gmail.com)
- */
-
-/* 二分查找最左一个元素 */
-function binarySearchLeftEdge(nums: number[], target: number): number {
-    let i = 0, j = nums.length - 1;  // 初始化双闭区间 [0, n-1]
-    while (i <= j) {
-        let m = Math.floor((i + j) / 2);  // 计算中点索引 m
-        if (nums[m] < target) {
-            i = m + 1;  // target 在区间 [m+1, j] 中
-        } else if (nums[m] > target) {
-            j = m - 1;  // target 在区间 [i, m-1] 中
-        } else {
-            j = m - 1;  // 首个小于 target 的元素在区间 [i, m-1] 中
-        }
-    }
-    if (i === nums.length || nums[i] != target) {
-        return -1;  // 未找到目标元素，返回 -1
-    }
-    return i;
-}
-
-/* 二分查找最右一个元素 */
-function binarySearchRightEdge(nums: number[], target: number): number {
-    let i = 0, j = nums.length - 1;  // 初始化双闭区间 [0, n-1]
-    while (i <= j) {
-        let m = Math.floor((i + j) / 2);  // 计算中点索引 m
-        if (nums[m] < target) {
-            i = m + 1;  // target 在区间 [m+1, j] 中
-        } else if (nums[m] > target) {
-            j = m - 1;  // target 在区间 [i, m-1] 中
-        } else {
-            i = m + 1;  // 首个大于 target 的元素在区间 [m+1, j] 中
-        }
-    }
-    if (j < 0 || nums[j] != target) {
-        return -1;  // 未找到目标元素，返回 -1
-    }
-    return j;
-}
-
-/* Driver Code */
-let target: number = 6;
-const nums: number[] = [1, 3, 6, 6, 6, 6, 6, 10, 12, 15];
-
-// 二分查找最左一个元素
-let index_left: number = binarySearchLeftEdge(nums, target);
-console.log("数组中最左一个元素 6 的索引 = ", index_left);
-
-// 二分查找最右一个元素
-let index_right: number = binarySearchRightEdge(nums, target);
-console.log("数组中最右一个元素 6 的索引 = ", index_right);
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index 6de8cd60..7f348ec2 100644
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diff --git a/docs/chapter_searching/binary_search.assets/binary_search_step7.png b/docs/chapter_searching/binary_search.assets/binary_search_step7.png
index 1b8f0fe4..196f5b1e 100644
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diff --git a/docs/chapter_searching/binary_search.md b/docs/chapter_searching/binary_search.md
index a8991b64..11c8d879 100755
--- a/docs/chapter_searching/binary_search.md
+++ b/docs/chapter_searching/binary_search.md
@@ -6,7 +6,9 @@
 
     给定一个长度为 $n$ 的数组 `nums` ，元素按从小到大的顺序排列，数组不包含重复元素。请查找并返回元素 `target` 在该数组中的索引。若数组不包含该元素，则返回 $-1$ 。
 
-对于上述问题，我们先初始化指针 $i = 0$ 和 $j = n - 1$ ，分别指向数组首元素和尾元素，代表搜索区间 $[0, n - 1]$ 。其中，中括号表示“闭区间”，即包含边界值本身。
+![二分查找示例数据](binary_search.assets/binary_search_example.png)
+
+对于上述问题，我们先初始化指针 $i = 0$ 和 $j = n - 1$ ，分别指向数组首元素和尾元素，代表搜索区间 $[0, n - 1]$ 。请注意，中括号表示闭区间，其包含边界值本身。
 
 接下来，循环执行以下两个步骤：
 
@@ -18,9 +20,6 @@
 
 若数组不包含目标元素，搜索区间最终会缩小为空。此时返回 $-1$ 。
 
-=== "<0>"
-    ![二分查找步骤](binary_search.assets/binary_search_step0.png)
-
 === "<1>"
     ![binary_search_step1](binary_search.assets/binary_search_step1.png)
 
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deleted file mode 100644
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diff --git a/docs/chapter_searching/binary_search_edge.assets/binary_search_right_edge_by_left_edge.png b/docs/chapter_searching/binary_search_edge.assets/binary_search_right_edge_by_left_edge.png
new file mode 100644
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diff --git a/docs/chapter_searching/binary_search_edge.md b/docs/chapter_searching/binary_search_edge.md
index 9ac6addf..4641efa2 100644
--- a/docs/chapter_searching/binary_search_edge.md
+++ b/docs/chapter_searching/binary_search_edge.md
@@ -1,56 +1,19 @@
 # 二分查找边界
 
-在上一节中，题目规定数组中所有元素都是唯一的。如果目标元素在数组中多次出现，上节介绍的方法只能保证返回其中一个目标元素的索引，**而无法确定该索引的左边和右边还有多少目标元素**。
+## 查找左边界
 
 !!! question
 
-    给定一个长度为 $n$ 的有序数组 `nums` ，数组可能包含重复元素。请查找并返回元素 `target` 在数组中首次出现的索引。若数组中不包含该元素，则返回 $-1$ 。
+    给定一个长度为 $n$ 的有序数组 `nums` ，数组可能包含重复元素。请返回数组中最左一个元素 `target` 的索引。若数组中不包含该元素，则返回 $-1$ 。
 
-## 线性方法
+回忆二分查找插入点的方法，搜索完成后，$i$ 指向最左一个 `target` ，**因此查找插入点本质上是在查找最左一个 `target` 的索引**。
 
-为了查找数组中最左边的 `target` ，我们可以分为两步：
+考虑通过查找插入点的函数实现查找左边界。请注意，数组中可能不包含 `target` ，此时有两种可能：
 
-1. 进行二分查找，定位到任意一个 `target` 的索引，记为 $k$ 。
-2. 以索引 $k$ 为起始点，向左进行线性遍历，找到最左边的 `target` 返回即可。
+1. 插入点的索引 $i$ 越界；
+2. 元素 `nums[i]` 与 `target` 不相等；
 
-![线性查找最左边的元素](binary_search_edge.assets/binary_search_left_edge_naive.png)
-
-这个方法虽然有效，但由于包含线性查找，时间复杂度为 $O(n)$ ，当存在很多重复的 `target` 时效率较低。
-
-## 二分方法
-
-考虑仅使用二分查找解决该问题。整体算法流程不变，先计算中点索引 $m$ ，再判断 `target` 和 `nums[m]` 大小关系：
-
-- 当 `nums[m] < target` 或 `nums[m] > target` 时，说明还没有找到 `target` ，因此采取与上节代码相同的缩小区间操作，**从而使指针 $i$ 和 $j$ 向 `target` 靠近**。
-- 当 `nums[m] == target` 时，说明“小于 `target` 的元素”在区间 $[i, m - 1]$ 中，因此采用 $j = m - 1$ 来缩小区间，**从而使指针 $j$ 向小于 `target` 的元素靠近**。
-
-二分查找完成后，**$i$ 指向最左边的 `target` ，$j$ 指向首个小于 `target` 的元素**，因此返回索引 $i$ 即可。
-
-=== "<1>"
-    ![二分查找最左边元素的步骤](binary_search_edge.assets/binary_search_left_edge_step1.png)
-
-=== "<2>"
-    ![binary_search_left_edge_step2](binary_search_edge.assets/binary_search_left_edge_step2.png)
-
-=== "<3>"
-    ![binary_search_left_edge_step3](binary_search_edge.assets/binary_search_left_edge_step3.png)
-
-=== "<4>"
-    ![binary_search_left_edge_step4](binary_search_edge.assets/binary_search_left_edge_step4.png)
-
-=== "<5>"
-    ![binary_search_left_edge_step5](binary_search_edge.assets/binary_search_left_edge_step5.png)
-
-=== "<6>"
-    ![binary_search_left_edge_step6](binary_search_edge.assets/binary_search_left_edge_step6.png)
-
-=== "<7>"
-    ![binary_search_left_edge_step7](binary_search_edge.assets/binary_search_left_edge_step7.png)
-
-=== "<8>"
-    ![binary_search_left_edge_step8](binary_search_edge.assets/binary_search_left_edge_step8.png)
-
-注意，数组可能不包含目标元素 `target` 。因此在函数返回前，我们需要先判断 `nums[i]` 与 `target` 是否相等，以及索引 $i$ 是否越界。
+当遇到以上两种情况时，直接返回 $-1$ 即可。
 
 === "Java"
 
@@ -126,9 +89,19 @@
 
 ## 查找右边界
 
-类似地，我们也可以二分查找最右边的 `target` 。当 `nums[m] == target` 时，说明大于 `target` 的元素在区间 $[m + 1, j]$ 中，因此执行 `i = m + 1` ，**使得指针 $i$ 向大于 `target` 的元素靠近**。
+那么如何查找最右一个 `target` 呢？最直接的方式是修改代码，替换在 `nums[m] == target` 情况下的指针收缩操作。代码在此省略，有兴趣的同学可以自行实现。
 
-完成二分后，**$i$ 指向首个大于 `target` 的元素，$j$ 指向最右边的 `target`** ，因此返回索引 $j$ 即可。
+下面我们介绍两种更加取巧的方法。
+
+### 复用查找左边界
+
+实际上，我们可以利用查找最左元素的函数来查找最右元素，具体方法为：**将查找最右一个 `target` 转化为查找最左一个 `target + 1`**。
+
+查找完成后，指针 $i$ 指向最左一个 `target + 1`（如果存在），而 $j$ 指向最右一个 `target` ，**因此返回 $j$ 即可**。
+
+![将查找右边界转化为查找左边界](binary_search_edge.assets/binary_search_right_edge_by_left_edge.png)
+
+请注意，返回的插入点是 $i$ ，因此需要将其减 $1$ ，从而获得 $j$ 。
 
 === "Java"
 
@@ -202,10 +175,18 @@
     [class]{}-[func]{binary_search_right_edge}
     ```
 
-观察下图，搜索最右边元素时指针 $j$ 的作用与搜索最左边元素时指针 $i$ 的作用一致，反之亦然。也就是说，**搜索最左边元素和最右边元素的实现是镜像对称的**。
+### 转化为查找元素
 
-![查找最左边和最右边元素的对称性](binary_search_edge.assets/binary_search_left_right_edge.png)
+我们知道，当数组不包含 `target` 时，最后 $i$ , $j$ 会分别指向首个大于、小于 `target` 的元素。
 
-!!! tip
+根据上述结论，我们可以构造一个数组中不存在的元素，用于查找左右边界：
 
-    以上代码采取的都是“双闭区间”写法。有兴趣的读者可以自行实现“左闭右开”写法。
+- 查找最左一个 `target` ：可以转化为查找 `target - 0.5` ，并返回指针 $i$ 。
+- 查找最右一个 `target` ：可以转化为查找 `target + 0.5` ，并返回指针 $j$ 。
+
+![将查找边界转化为查找元素](binary_search_edge.assets/binary_search_edge_by_element.png)
+
+代码在此省略，值得注意的有：
+
+- 给定数组不包含小数，这意味着我们无需关心如何处理相等的情况。
+- 因为该方法引入了小数，所以需要将函数中的变量 `target` 改为浮点数类型。
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diff --git a/docs/chapter_searching/binary_search_insertion.md b/docs/chapter_searching/binary_search_insertion.md
new file mode 100644
index 00000000..31ca1935
--- /dev/null
+++ b/docs/chapter_searching/binary_search_insertion.md
@@ -0,0 +1,227 @@
+# 二分查找插入点
+
+二分查找不仅可用于搜索目标元素，还具有许多变种问题，比如搜索目标元素的插入位置。
+
+## 无重复元素的情况
+
+!!! question
+
+    给定一个长度为 $n$ 的有序数组 `nums` 和一个元素 `target` ，数组不存在重复元素。现将 `target` 插入到数组 `nums` 中，并保持其有序性。若数组中已存在元素 `target` ，则插入到其左方。请返回插入后 `target` 在数组中的索引。
+
+![二分查找插入点示例数据](binary_search_insertion.assets/binary_search_insertion_example.png)
+
+如果想要复用上节的二分查找代码，则需要回答以下两个问题。
+
+**问题一**：当数组中包含 `target` 时，插入点的索引是否是该元素的索引？
+
+题目要求将 `target` 插入到相等元素的左边，这意味着新插入的 `target` 替换了原来 `target` 的位置。也就是说，**当数组包含 `target` 时，插入点的索引就是该 `target` 的索引**。
+
+**问题二**：当数组中不存在 `target` 时，插入点是哪个元素的索引？
+
+进一步思考二分查找过程：当 `nums[m] < target` 时 $i$ 移动，这意味着指针 $i$ 在向大于等于 `target` 的元素靠近。同理，指针 $j$ 始终在向小于等于 `target` 的元素靠近。
+
+因此二分结束时一定有：$i$ 指向首个大于 `target` 的元素，$j$ 指向首个小于 `target` 的元素。**易得当数组不包含 `target` 时，插入索引为 $i$** 。
+
+=== "Java"
+
+    ```java title="binary_search_insertion.java"
+    [class]{binary_search_insertion}-[func]{binarySearchInsertionSimple}
+    ```
+
+=== "C++"
+
+    ```cpp title="binary_search_insertion.cpp"
+    [class]{}-[func]{binarySearchInsertionSimple}
+    ```
+
+=== "Python"
+
+    ```python title="binary_search_insertion.py"
+    [class]{}-[func]{binary_search_insertion_simple}
+    ```
+
+=== "Go"
+
+    ```go title="binary_search_insertion.go"
+    [class]{}-[func]{binarySearchInsertionSimple}
+    ```
+
+=== "JS"
+
+    ```javascript title="binary_search_insertion.js"
+    [class]{}-[func]{binarySearchInsertionSimple}
+    ```
+
+=== "TS"
+
+    ```typescript title="binary_search_insertion.ts"
+    [class]{}-[func]{binarySearchInsertionSimple}
+    ```
+
+=== "C"
+
+    ```c title="binary_search_insertion.c"
+    [class]{}-[func]{binarySearchInsertionSimple}
+    ```
+
+=== "C#"
+
+    ```csharp title="binary_search_insertion.cs"
+    [class]{binary_search_insertion}-[func]{binarySearchInsertionSimple}
+    ```
+
+=== "Swift"
+
+    ```swift title="binary_search_insertion.swift"
+    [class]{}-[func]{binarySearchInsertionSimple}
+    ```
+
+=== "Zig"
+
+    ```zig title="binary_search_insertion.zig"
+    [class]{}-[func]{binarySearchInsertionSimple}
+    ```
+
+=== "Dart"
+
+    ```dart title="binary_search_insertion.dart"
+    [class]{}-[func]{binarySearchInsertionSimple}
+    ```
+
+=== "Rust"
+
+    ```rust title="binary_search_insertion.rs"
+    [class]{}-[func]{binary_search_insertion}
+    ```
+
+## 存在重复元素的情况
+
+!!! question
+
+    在上一题的基础上，规定数组可能包含重复元素，其余不变。
+
+假设数组中存在多个 `target` ，则普通二分查找只能返回其中一个 `target` 的索引，**而无法确定该元素的左边和右边还有多少 `target`**。
+
+题目要求将目标元素插入到最左边，**所以我们需要查找数组中最左一个 `target` 的索引**。初步考虑通过以下两步实现：
+
+1. 执行二分查找，得到任意一个 `target` 的索引，记为 $k$ 。
+2. 从索引 $k$ 开始，向左进行线性遍历，当找到最左边的 `target` 时返回。
+
+![线性查找重复元素的插入点](binary_search_insertion.assets/binary_search_insertion_naive.png)
+
+此方法虽然可用，但其包含线性查找，因此时间复杂度为 $O(n)$ 。当数组中存在很多重复的 `target` 时，该方法效率很低。
+
+现考虑修改二分查找代码。整体流程不变，每轮先计算中点索引 $m$ ，再判断 `target` 和 `nums[m]` 大小关系：
+
+1. 当 `nums[m] < target` 或 `nums[m] > target` 时，说明还没有找到 `target` ，因此采用普通二分查找的缩小区间操作，**从而使指针 $i$ 和 $j$ 向 `target` 靠近**。
+2. 当 `nums[m] == target` 时，说明小于 `target` 的元素在区间 $[i, m - 1]$ 中，因此采用 $j = m - 1$ 来缩小区间，**从而使指针 $j$ 向小于 `target` 的元素靠近**。
+
+循环完成后，$i$ 指向最左边的 `target` ，$j$ 指向首个小于 `target` 的元素，**因此索引 $i$ 就是插入点**。
+
+=== "<1>"
+    ![二分查找重复元素的插入点的步骤](binary_search_insertion.assets/binary_search_insertion_step1.png)
+
+=== "<2>"
+    ![binary_search_insertion_step2](binary_search_insertion.assets/binary_search_insertion_step2.png)
+
+=== "<3>"
+    ![binary_search_insertion_step3](binary_search_insertion.assets/binary_search_insertion_step3.png)
+
+=== "<4>"
+    ![binary_search_insertion_step4](binary_search_insertion.assets/binary_search_insertion_step4.png)
+
+=== "<5>"
+    ![binary_search_insertion_step5](binary_search_insertion.assets/binary_search_insertion_step5.png)
+
+=== "<6>"
+    ![binary_search_insertion_step6](binary_search_insertion.assets/binary_search_insertion_step6.png)
+
+=== "<7>"
+    ![binary_search_insertion_step7](binary_search_insertion.assets/binary_search_insertion_step7.png)
+
+=== "<8>"
+    ![binary_search_insertion_step8](binary_search_insertion.assets/binary_search_insertion_step8.png)
+
+观察以下代码，判断分支 `nums[m] > target` 和 `nums[m] == target` 的操作相同，因此两者可以合并。
+
+即便如此，我们仍然可以将判断条件保持展开，因为其逻辑更加清晰、可读性更好。
+
+=== "Java"
+
+    ```java title="binary_search_insertion.java"
+    [class]{binary_search_insertion}-[func]{binarySearchInsertion}
+    ```
+
+=== "C++"
+
+    ```cpp title="binary_search_insertion.cpp"
+    [class]{}-[func]{binarySearchInsertion}
+    ```
+
+=== "Python"
+
+    ```python title="binary_search_insertion.py"
+    [class]{}-[func]{binary_search_insertion}
+    ```
+
+=== "Go"
+
+    ```go title="binary_search_insertion.go"
+    [class]{}-[func]{binarySearchInsertion}
+    ```
+
+=== "JS"
+
+    ```javascript title="binary_search_insertion.js"
+    [class]{}-[func]{binarySearchInsertion}
+    ```
+
+=== "TS"
+
+    ```typescript title="binary_search_insertion.ts"
+    [class]{}-[func]{binarySearchInsertion}
+    ```
+
+=== "C"
+
+    ```c title="binary_search_insertion.c"
+    [class]{}-[func]{binarySearchInsertion}
+    ```
+
+=== "C#"
+
+    ```csharp title="binary_search_insertion.cs"
+    [class]{binary_search_insertion}-[func]{binarySearchInsertion}
+    ```
+
+=== "Swift"
+
+    ```swift title="binary_search_insertion.swift"
+    [class]{}-[func]{binarySearchInsertion}
+    ```
+
+=== "Zig"
+
+    ```zig title="binary_search_insertion.zig"
+    [class]{}-[func]{binarySearchInsertion}
+    ```
+
+=== "Dart"
+
+    ```dart title="binary_search_insertion.dart"
+    [class]{}-[func]{binarySearchInsertion}
+    ```
+
+=== "Rust"
+
+    ```rust title="binary_search_insertion.rs"
+    [class]{}-[func]{binary_search_insertion}
+    ```
+
+!!! tip
+
+    本节的代码都是“双闭区间”写法。有兴趣的读者可以自行实现“左闭右开”写法。
+
+总的来看，二分查找无非就是给指针 $i$ , $j$ 分别设定搜索目标，目标可能是一个具体的元素（例如 `target` ），也可能是一个元素范围（例如小于 `target` 的元素）。
+
+在不断的循环二分中，指针 $i$ , $j$ 都逐渐逼近预先设定的目标。最终，它们或是成功找到答案，或是越过边界后停止。
diff --git a/mkdocs.yml b/mkdocs.yml
index c7a6d3eb..0ee62480 100644
--- a/mkdocs.yml
+++ b/mkdocs.yml
@@ -207,10 +207,13 @@ nav:
     # [icon: material/text-search]
     - chapter_searching/index.md
     - 10.1. &nbsp; 二分查找: chapter_searching/binary_search.md
-    - 10.2. &nbsp; 二分查找边界: chapter_searching/binary_search_edge.md 
-    - 10.3. &nbsp; 哈希优化策略: chapter_searching/replace_linear_by_hashing.md
-    - 10.4. &nbsp; 重识搜索算法: chapter_searching/searching_algorithm_revisited.md
-    - 10.5. &nbsp; 小结: chapter_searching/summary.md
+    # [status: new]
+    - 10.2. &nbsp; 二分查找插入点: chapter_searching/binary_search_insertion.md
+    # [status: new]
+    - 10.3. &nbsp; 二分查找边界: chapter_searching/binary_search_edge.md
+    - 10.4. &nbsp; 哈希优化策略: chapter_searching/replace_linear_by_hashing.md
+    - 10.5. &nbsp; 重识搜索算法: chapter_searching/searching_algorithm_revisited.md
+    - 10.6. &nbsp; 小结: chapter_searching/summary.md
   - 11. &nbsp; 排序:
     # [icon: material/sort-ascending]
     - chapter_sorting/index.md