Triángulo de Reuleaux más grande dentro de un cuadrado que está inscrito dentro de un triángulo de ángulo recto

Aquí se da un triángulo rectángulo con altura l , base b e hipotenusa h , que inscribe un cuadrado que a su vez inscribe un triángulo reuleaux. La tarea es encontrar el área máxima posible de este triángulo de Reuleaux.
Ejemplos: 
 

Input: l = 5, b = 12, h = 13
Output: 8.77914

Input: l = 3, b = 4, h = 5
Output: 2.07116

Enfoque : sabemos que el lado del cuadrado inscrito dentro de un triángulo rectángulo es a = (l*b)/(l+b) , consulte Área del cuadrado más grande que cabe en un triángulo rectángulo . 
Además, en el triángulo de Reuleaux, x = a . 
Entonces, x = (l*b)/(l+b) . 
Entonces, el área del triángulo de Reuleaux es A = 0.70477*x^2 = 0.70477*((l*b)/(l+b))^2 .
A continuación se muestra la implementación del enfoque anterior: 
 

// C++ Program to find the biggest Reuleaux triangle
// inscribed within in a square which in turn
// is inscribed within a circle
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the biggest reuleaux triangle
float Area(float l, float b, float h)
{
 
    // the height or base or hypotenuse
    // cannot be negative
    if (l < 0 || b < 0 || h < 0)
        return -1;
 
    // height of the reuleaux triangle
    float x = (l * b) / (l + b);
 
    // area of the reuleaux triangle
    float A = 0.70477 * pow(x, 2);
 
    return A;
}
 
// Driver code
int main()
{
    float l = 5, b = 12, h = 13;
    cout << Area(l, b, h) << endl;
 
    return 0;
}

// Java Program to find the biggest Reuleaux triangle
// inscribed within in a square which in turn
// is inscribed within a circle
import java.util.*;
import java.text.DecimalFormat;
 
class GFG
{
 
// Function to find the biggest reuleaux triangle
static double Area(double l, double b, double h)
{
 
    // the height or base or hypotenuse
    // cannot be negative
    if (l < 0 || b < 0 || h < 0)
        return -1;
 
    // height of the reuleaux triangle
    double x = (l * b) / (l + b);
 
    // area of the reuleaux triangle
    double A = 0.70477 * Math.pow(x, 2);
 
    return A;
}
 
// Driver code
public static void main(String args[])
{
    double l = 5, b = 12, h = 13;
    DecimalFormat df = new DecimalFormat("#,###,##0.00000");
    System.out.println(df.format(Area(l, b, h)));
}
}
 
// This code is contributed by
// Shashank_Sharma

# Python3 Program to find the biggest
# Reuleaux triangle inscribed within
# in a square which in turn is inscribed
# within a circle
import math as mt
 
# Function to find the biggest
# reuleaux triangle
def Area(l, b, h):
 
    # the height or base or hypotenuse
    # cannot be negative
    if (l < 0 or b < 0 or h < 0):
        return -1
 
    # height of the reuleaux triangle
    x = (l * b) /(l + b)
 
    # area of the reuleaux triangle
    A = 0.70477 * pow(x, 2)
 
    return A
 
# Driver code
l, b, h = 5, 12, 13
print(Area(l, b, h))
 
# This code is contributed by
# Mohit kumar 29

// C# Program to find the biggest Reuleaux triangle
// inscribed within in a square which in turn
// is inscribed within a circle
using System;
 
class GFG
{
 
// Function to find the biggest reuleaux triangle
static double Area(double l, double b, double h)
{
 
    // the height or base or hypotenuse
    // cannot be negative
    if (l < 0 || b < 0 || h < 0)
        return -1;
 
    // height of the reuleaux triangle
    double x = (l * b) / (l + b);
 
    // area of the reuleaux triangle
    double A = 0.70477 * Math.Pow(x, 2);
 
    return A;
}
 
// Driver code
public static void Main()
{
    double l = 5, b = 12, h = 13;
    Console.WriteLine((Area(l, b, h)));
}
}
 
// This code is contributed by
// Mukul Singh

<?php
// PHP Program to find the biggest
// Reuleaux triangle inscribed within
// in a square which in turn is
// inscribed within a circle
 
// Function to find the biggest
// reuleaux triangle
function Area($l, $b, $h)
{
 
    // the height or base or hypotenuse
    // cannot be negative
    if ($l < 0 or $b < 0 or $h < 0)
        return -1;
 
    // height of the reuleaux triangle
    $x = ($l * $b) / ($l + $b);
 
    // area of the reuleaux triangle
    $A = 0.70477 * pow($x, 2);
 
    return $A;
}
 
// Driver code
$l = 5; $b = 12; $h = 13;
echo Area($l, $b, $h);
 
// This code is contributed by
// anuj_67
?>

<script>
 
// Javascript Program to find the biggest Reuleaux triangle
// inscribed within in a square which in turn
// is inscribed within a circle
 
 
// Function to find the biggest reuleaux triangle
function Area(l,b,h)
{
 
    // the height or base or hypotenuse
    // cannot be negative
    if (l < 0 || b < 0 || h < 0)
        return -1;
 
    // height of the reuleaux triangle
    let x = (l * b) / (l + b);
 
    // area of the reuleaux triangle
    let A = 0.70477 * Math.pow(x, 2);
 
    return A;
}
 
// Driver code
let l = 5, b = 12, h = 13;
  
    document.write( Area(l,b,h).toFixed(5));
 
// This code contributed by Rajput-Ji
 
</script>

8.77914

Complejidad de tiempo: O(1) 

Espacio Auxiliar: O(1)