Área del mayor triángulo isósceles que se puede inscribir en una Elipse cuyo vértice coincide con un extremo del eje mayor

Dada una elipse con la mitad de los ejes mayor y menor de longitud A y B , la tarea es encontrar el área del triángulo isósceles más grande que se puede inscribir en la elipse cuyo vértice coincide con un extremo del eje mayor.

Ejemplos:

Entrada: A = 1, B = 2
Salida: 2,598
Explicación:
Área del triángulo isósceles = ((3 * √3) * A * B) / 4.
Por lo tanto, área = 2,598.

Entrada: A = 2, B = 3
Salida: 7.794

Planteamiento: La idea se basa en la siguiente fórmula matemática:

Prueba: 

Considerando el triángulo APB, 
Área de APB = AB * PQ = (1 / 2) * A * B * (2 sin∅ – sin2∅)

Derivando : 
d(area(APB))/d∅ = ab ( cos∅ – cos2∅)

Igualando la derivada a cero: 
d(area(APB))/d∅ = 0 
cos∅ = – (1 / 2) 
∅ = 2PI / 3

Por lo tanto, área de APB = (3√3) * A * B / 4

A continuación se muestra la implementación del enfoque anterior:

// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to calculate area
// of the isosceles triangle
void triangleArea(float a, float b)
{
    // If a and b are negative
    if (a < 0 || b < 0) {
        cout << -1;
        return;
    }
 
    // Stores the area of the triangle
    float area = (3 * sqrt(3) * a * b) / (4);
 
    // Print the area
    cout << area;
}
 
// Driver code
int main()
{
    // Given value of a & b
    float a = 1, b = 2;
 
    // Function call to find the
    // area of the isosceles triangle
    triangleArea(a, b);
 
    return 0;
}

// Java program for the above approach
import java.util.*;
 
class GFG
{
 
// Function to calculate area
// of the isosceles triangle
static void triangleArea(float a, float b)
{
   
    // If a and b are negative
    if (a < 0 || b < 0) {
        System.out.println(-1);
        return;
    }
 
    // Stores the area of the triangle
    float area = (3 * (float)Math.sqrt(3) * a * b) / (4);
 
    // Print the area
    System.out.println(area);
}
 
// Driver Code
public static void main(String[] args)
{
   
    // Given value of a & b
    float a = 1, b = 2;
 
    // Function call to find the
    // area of the isosceles triangle
    triangleArea(a, b);
}
}
 
// This code is contributed by sanjoy_62.

# Python 3 program for the above approach
from math import sqrt
 
# Function to calculate area
# of the isosceles triangle
def triangleArea(a, b):
   
    # If a and b are negative
    if (a < 0 or b < 0):
        print(-1)
        return
 
    # Stores the area of the triangle
    area = (3 * sqrt(3) * a * b) / (4);
 
    # Print the area
    print("{:.5f}".format(area))
 
# Driver code
if __name__ == '__main__':
   
    # Given value of a & b
    a = 1
    b = 2
     
    # Function call to find the
    # area of the isosceles triangle
    triangleArea(a, b)
     
    # This code is contributed by SURENDRA_GANGWAR.

// C# program for the above approach
 
using System;
public class GFG
{
 
  // Function to calculate area
  // of the isosceles triangle
  static void triangleArea(float a, float b)
  {
 
    // If a and b are negative
    if (a < 0 || b < 0) {
      Console.WriteLine(-1);
      return;
    }
 
    // Stores the area of the triangle
    float area = (3 * (float)Math.Sqrt(3) * a * b) / (4);
 
    // Print the area
    Console.WriteLine(area);
  }
 
  // Driver Code
  public static void Main(string[] args)
  {
 
    // Given value of a & b
    float a = 1, b = 2;
 
    // Function call to find the
    // area of the isosceles triangle
    triangleArea(a, b);
  }
}
 
// This code is contributed by AnkThon

<script>
 
// Javascript program for the above approach
 
// Function to calculate area
// of the isosceles triangle
function triangleArea(a, b)
{
     
    // If a and b are negative
    if (a < 0 || b < 0)
    {
        document.write(-1);
        return;
    }
 
    // Stores the area of the triangle
    var area = (3 *  Math.sqrt(3) * a * b) / (4);
 
    // Print the area
    document.write(area.toFixed(5));
}
 
// Driver Code
 
// Given value of a & b
var a = 1, b = 2;
 
// Function call to find the
// area of the isosceles triangle
triangleArea(a, b);
 
// This code is contributed by todaysgaurav
 
</script>

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