Maximize area of triangle formed by points on sides of given rectangle

Given a rectangle [(x1, y1), (x2, y2)] denoting the coordinates of bottom-left corner and top-right corner whose sides are parallel to coordinates axes and N points on its perimeter (at least one on each side). The task is to maximize the area of a triangle formed by these points.

Examples:

Input: rectangle[][]= {{0, 0}, {6, 6}},
coordinates[][] = {{0, 2}, {0, 3}, {0, 5}, {2, 0}, {3, 0}, {6, 0}, {6, 4}, {1, 6}, {6, 6}}
Output: 18
Explanation: Refer to the image below for explanation.

Approach: For finding the maximum area triangle by coordinates on a given rectangle, find the coordinates on each side which are most distant apart. So, suppose there are four sides a, b, c, d where a and c being the length of the rectangle and b, d being the breadth. Now the maximum area will be
MAX ( length * ( distance between farthest coordinates on either of breadth) / 2, breadth * ( distance between farthest coordinates on either of length) / 2 ).
Follow the steps below to solve the given problem.

Calculate the length = abs(x2 - x1) and breadth = abs(y2 - y1)
Find the coordinates which are farthest from each other on each side.
Use the above-mentioned formula to calculate the area.
Return the area found.

Below is the implementation of the above approach.

C++

// C++ program for above approach
#include <bits/stdc++.h>
#include <iostream>
using namespace std;

// To find the maximum area of triangle
void maxTriangleArea(int rectangle[2][2],
                     int coordinates[][2],
                     int numberOfCoordinates)
{

    int l1min = INT_MAX, l2min = INT_MAX,
        l1max = INT_MIN, l2max = INT_MIN,
        b1min = INT_MAX, b1max = INT_MIN,
        b2min = INT_MAX, b2max = INT_MIN;

    int l1Ycoordinate = rectangle[0][1];
    int l2Ycoordinate = rectangle[1][1];

    int b1Xcoordinate = rectangle[0][0];
    int b2Xcoordinate = rectangle[1][0];

    // Always consider side parallel
    // to x-axis as length and
    // side parallel to y-axis as breadth
    for (int i = 0; i < numberOfCoordinates;
         i++) {
        coordinates[i][1];

        // coordinate on l1
        if (coordinates[i][1] == l1Ycoordinate) {
            l1min = min(l1min,
                        coordinates[i][0]);
            l1max = max(l1max,
                        coordinates[i][0]);
        }

        // Coordinate on l2
        if (coordinates[i][1] == l2Ycoordinate) {
            l2min = min(l2min,
                        coordinates[i][0]);
            l2max = max(l2max,
                        coordinates[i][0]);
        }

        // Coordinate on b1
        if (coordinates[i][0] == b1Xcoordinate) {
            b1min = min(b1min,
                        coordinates[i][1]);
            b1max = max(b1max,
                        coordinates[i][1]);
        }

        // Coordinate on b2
        if (coordinates[i][0] == b2Xcoordinate) {
            b2min = min(b2min,
                        coordinates[i][1]);
            b2max = max(b2max,
                        coordinates[i][1]);
        }
    }

    // Find maximum possible distance
    // on length
    int maxOfLength = max(abs(l1max - l1min),
                          abs(l2max - l2min));

    // Find maximum possible distance
    // on breadth
    int maxofBreadth = max(abs(b1max - b1min),
                           abs(b2max - b2min));

    // Calculate result base * height / 2
    float result
        = max((maxofBreadth
               * (abs(rectangle[0][0]
                      - rectangle[1][0]))),
              (maxOfLength
               * (abs(rectangle[0][1]
                      - rectangle[1][1]))))
          / 2.0;

    // Print the result
    cout << result;
}

// Driver Code
int main()
{
    // Rectangle with x1, y1 and x2, y2
    int rectangle[2][2] = { { 0, 0 },
                            { 6, 6 } };

    // Coordinates on sides of given rectangle
    int coordinates[9][2]
        = { { 0, 2 }, { 0, 3 }, { 0, 5 },
            { 2, 0 }, { 3, 0 }, { 6, 0 },
            { 6, 4 }, { 1, 6 }, { 6, 6 } };

    int numberOfCoordinates
        = sizeof(coordinates) / sizeof(coordinates[0]);

    maxTriangleArea(rectangle, coordinates,
                    numberOfCoordinates);
    return 0;
}

Java

// Java program for the above approach
import java.io.*;
import java.lang.*;
import java.util.*;

class GFG {

  // To find the maximum area of triangle
  static void maxTriangleArea(int[ ][ ] rectangle,
                              int[ ][ ] coordinates,
                              int numberOfCoordinates)
  {

    int l1min = Integer.MAX_VALUE, l2min = Integer.MAX_VALUE,
    l1max = Integer.MIN_VALUE, l2max = Integer.MIN_VALUE,
    b1min = Integer.MAX_VALUE, b1max = Integer.MIN_VALUE,
    b2min = Integer.MAX_VALUE, b2max = Integer.MIN_VALUE;

    int l1Ycoordinate = rectangle[0][1];
    int l2Ycoordinate = rectangle[1][1];

    int b1Xcoordinate = rectangle[0][0];
    int b2Xcoordinate = rectangle[1][0];

    // Always consider side parallel
    // to x-axis as length and
    // side parallel to y-axis as breadth
    for (int i = 0; i < numberOfCoordinates; i++) {

      // coordinate on l1
      if (coordinates[i][1] == l1Ycoordinate) {
        l1min = Math.min(l1min,
                         coordinates[i][0]);
        l1max = Math.max(l1max,
                         coordinates[i][0]);
      }

      // Coordinate on l2
      if (coordinates[i][1] == l2Ycoordinate) {
        l2min = Math.min(l2min,
                         coordinates[i][0]);
        l2max = Math.max(l2max,
                         coordinates[i][0]);
      }

      // Coordinate on b1
      if (coordinates[i][0] == b1Xcoordinate) {
        b1min = Math.min(b1min,
                         coordinates[i][1]);
        b1max = Math.max(b1max,
                         coordinates[i][1]);
      }

      // Coordinate on b2
      if (coordinates[i][0] == b2Xcoordinate) {
        b2min = Math.min(b2min,
                         coordinates[i][1]);
        b2max = Math.max(b2max,
                         coordinates[i][1]);
      }
    }

    // Find maximum possible distance
    // on length
    int maxOfLength = Math.max(Math.abs(l1max - l1min),
                               Math.abs(l2max - l2min));

    // Find maximum possible distance
    // on breadth
    int maxofBreadth = Math.max(Math.abs(b1max - b1min),
                                Math.abs(b2max - b2min));

    // Calculate result base * height / 2
    int result
      = Math.max((maxofBreadth
                  * (Math.abs(rectangle[0][0]
                              - rectangle[1][0]))),
                 (maxOfLength
                  * (Math.abs(rectangle[0][1]
                              - rectangle[1][1]))))
      / 2;

    // Print the result
    System.out.print(result);
  }

  // Driver Code
  public static void main (String[] args)
  {
    
    // Rectangle with x1, y1 and x2, y2
    int[ ][ ] rectangle = { { 0, 0 },
                           { 6, 6 } };

    // Coordinates on sides of given rectangle
    int[ ][ ] coordinates
      = { { 0, 2 }, { 0, 3 }, { 0, 5 },
         { 2, 0 }, { 3, 0 }, { 6, 0 },
         { 6, 4 }, { 1, 6 }, { 6, 6 } };

    int numberOfCoordinates
      = coordinates.length;

    maxTriangleArea(rectangle, coordinates,
                    numberOfCoordinates);
  }
}

// This code is contributed by hrithikgarg03188.

Python3

# Python code for the above approach 

# To find the maximum area of triangle
def maxTriangleArea(rectangle, coordinates, numberOfCoordinates):

    l1min = 10 ** 9
    l2min = 10 ** 9
    l1max = 10 ** -9
    l2max = 10 ** -9
    b1min = 10 ** 9
    b1max = 10 ** -9
    b2min = 10 ** 9
    b2max = 10 ** -9

    l1Ycoordinate = rectangle[0][1];
    l2Ycoordinate = rectangle[1][1];

    b1Xcoordinate = rectangle[0][0];
    b2Xcoordinate = rectangle[1][0];

    # Always consider side parallel
    # to x-axis as length and
    # side parallel to y-axis as breadth
    for i in range(numberOfCoordinates):
        coordinates[i][1];

        # coordinate on l1
        if (coordinates[i][1] == l1Ycoordinate) :
            l1min = min(l1min, coordinates[i][0]);
            l1max = max(l1max, coordinates[i][0]);
        

        # Coordinate on l2
        if (coordinates[i][1] == l2Ycoordinate):
            l2min = min(l2min, coordinates[i][0]);
            l2max = max(l2max, coordinates[i][0]);


        # Coordinate on b1
        if (coordinates[i][0] == b1Xcoordinate):
            b1min = min(b1min, coordinates[i][1]);
            b1max = max(b1max, coordinates[i][1]);
        

        # Coordinate on b2
        if (coordinates[i][0] == b2Xcoordinate):
            b2min = min(b2min, coordinates[i][1]);
            b2max = max(b2max, coordinates[i][1]);
        

    # Find maximum possible distance
    # on length
    maxOfLength = max(abs(l1max - l1min), abs(l2max - l2min));

    # Find maximum possible distance
    # on breadth
    maxofBreadth = max(abs(b1max - b1min), abs(b2max - b2min));

    # Calculate result base * height / 2
    result = max((maxofBreadth * (abs(rectangle[0][0] - rectangle[1][0]))),
            (maxOfLength * (abs(rectangle[0][1] - rectangle[1][1])))) / 2.0;

    # Print the result
    print(int(result));


# Driver Code

# Rectangle with x1, y1 and x2, y2
rectangle = [[0, 0],[6, 6]];

# Coordinates on sides of given rectangle
coordinates = [[0, 2], [0, 3], [0, 5],
    [2, 0], [3, 0], [6, 0],
    [6, 4], [1, 6], [6, 6]];

numberOfCoordinates = len(coordinates)

maxTriangleArea(rectangle, coordinates, numberOfCoordinates);

# This code is contributed by gfgking

// C# program for the above approach
using System;
class GFG {

  // To find the maximum area of triangle
  static void maxTriangleArea(int[,] rectangle,
                              int[,] coordinates,
                              int numberOfCoordinates)
  {

    int l1min = Int32.MaxValue, l2min = Int32.MaxValue,
    l1max = Int32.MinValue, l2max = Int32.MinValue,
    b1min = Int32.MaxValue, b1max = Int32.MinValue,
    b2min = Int32.MaxValue, b2max = Int32.MinValue;

    int l1Ycoordinate = rectangle[0,1];
    int l2Ycoordinate = rectangle[1,1];

    int b1Xcoordinate = rectangle[0,0];
    int b2Xcoordinate = rectangle[1,0];

    // Always consider side parallel
    // to x-axis as length and
    // side parallel to y-axis as breadth
    for (int i = 0; i < numberOfCoordinates; i++) {

      // coordinate on l1
      if (coordinates[i,1] == l1Ycoordinate) {
        l1min = Math.Min(l1min,
                         coordinates[i,0]);
        l1max = Math.Max(l1max,
                         coordinates[i,0]);
      }

      // Coordinate on l2
      if (coordinates[i,1] == l2Ycoordinate) {
        l2min = Math.Min(l2min,
                         coordinates[i,0]);
        l2max = Math.Max(l2max,
                         coordinates[i,0]);
      }

      // Coordinate on b1
      if (coordinates[i,0] == b1Xcoordinate) {
        b1min = Math.Min(b1min,
                         coordinates[i,1]);
        b1max = Math.Max(b1max,
                         coordinates[i,1]);
      }

      // Coordinate on b2
      if (coordinates[i,0] == b2Xcoordinate) {
        b2min = Math.Min(b2min,
                         coordinates[i,1]);
        b2max = Math.Max(b2max,
                         coordinates[i,1]);
      }
    }

    // Find maximum possible distance
    // on length
    int maxOfLength = Math.Max(Math.Abs(l1max - l1min),
                               Math.Abs(l2max - l2min));

    // Find maximum possible distance
    // on breadth
    int maxofBreadth = Math.Max(Math.Abs(b1max - b1min),
                                Math.Abs(b2max - b2min));

    // Calculate result base * height / 2
    int result
      = Math.Max((maxofBreadth
                  * (Math.Abs(rectangle[0,0]
                              - rectangle[1,0]))),
                 (maxOfLength
                  * (Math.Abs(rectangle[0,1]
                              - rectangle[1,1]))))
      / 2;

    // Print the result
    Console.Write(result);
  }

  // Driver Code
  public static void Main ()
  {

    // Rectangle with x1, y1 and x2, y2
    int[,] rectangle = { { 0, 0 },
                        { 6, 6 } };

    // Coordinates on sides of given rectangle
    int[,] coordinates
      = { { 0, 2 }, { 0, 3 }, { 0, 5 },
         { 2, 0 }, { 3, 0 }, { 6, 0 },
         { 6, 4 }, { 1, 6 }, { 6, 6 } };

    int numberOfCoordinates
      = coordinates.GetLength(0);

    maxTriangleArea(rectangle, coordinates,
                    numberOfCoordinates);
  }
}

// This code is contributed by Samim Hossain Mondal.

JavaScript

 <script>
        // JavaScript code for the above approach 


        // To find the maximum area of triangle
        function maxTriangleArea(rectangle,
            coordinates,
            numberOfCoordinates) {

            let l1min = Number.MAX_VALUE, l2min = Number.MAX_VALUE,
                l1max = Number.MIN_VALUE, l2max = Number.MIN_VALUE,
                b1min = Number.MAX_VALUE, b1max = Number.MIN_VALUE,
                b2min = Number.MAX_VALUE, b2max = Number.MIN_VALUE;

            let l1Ycoordinate = rectangle[0][1];
            let l2Ycoordinate = rectangle[1][1];

            let b1Xcoordinate = rectangle[0][0];
            let b2Xcoordinate = rectangle[1][0];

            // Always consider side parallel
            // to x-axis as length and
            // side parallel to y-axis as breadth
            for (let i = 0; i < numberOfCoordinates;
                i++) {
                coordinates[i][1];

                // coordinate on l1
                if (coordinates[i][1] == l1Ycoordinate) {
                    l1min = Math.min(l1min,
                        coordinates[i][0]);
                    l1max = Math.max(l1max,
                        coordinates[i][0]);
                }

                // Coordinate on l2
                if (coordinates[i][1] == l2Ycoordinate) {
                    l2min = Math.min(l2min,
                        coordinates[i][0]);
                    l2max = Math.max(l2max,
                        coordinates[i][0]);
                }

                // Coordinate on b1
                if (coordinates[i][0] == b1Xcoordinate) {
                    b1min = Math.min(b1min,
                        coordinates[i][1]);
                    b1max = Math.max(b1max,
                        coordinates[i][1]);
                }

                // Coordinate on b2
                if (coordinates[i][0] == b2Xcoordinate) {
                    b2min = Math.min(b2min,
                        coordinates[i][1]);
                    b2max = Math.max(b2max,
                        coordinates[i][1]);
                }
            }

            // Find maximum possible distance
            // on length
            let maxOfLength = Math.max(Math.abs(l1max - l1min),
                Math.abs(l2max - l2min));

            // Find maximum possible distance
            // on breadth
            let maxofBreadth = Math.max(Math.abs(b1max - b1min),
                Math.abs(b2max - b2min));

            // Calculate result base * height / 2
            let result
                = Math.max((maxofBreadth
                    * (Math.abs(rectangle[0][0]
                        - rectangle[1][0]))),
                    (maxOfLength
                        * (Math.abs(rectangle[0][1]
                            - rectangle[1][1]))))
                / 2.0;

            // Print the result
            document.write(result);
        }

        // Driver Code

        // Rectangle with x1, y1 and x2, y2
        let rectangle = [[0, 0],
        [6, 6]];

        // Coordinates on sides of given rectangle
        let coordinates
            = [[0, 2], [0, 3], [0, 5],
            [2, 0], [3, 0], [6, 0],
            [6, 4], [1, 6], [6, 6]];

        let numberOfCoordinates
            = coordinates.length;

        maxTriangleArea(rectangle, coordinates,
            numberOfCoordinates);

       // This code is contributed by Potta Lokesh
    </script>

Output

Time complexity: O(N), Where N is the number of coordinates given.
Auxiliary Space: O(1)

Maximize area of triangle formed by points on sides of given rectangle

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