Dados tres pares de números enteros A(x, y) , B(x, y) y C(x, y) , que representan las coordenadas de un triángulo rectángulo , la tarea es encontrar la distancia entre el ortocentro y el circuncentro .
Ejemplos:
Entrada: A = {0, 0}, B = {5, 0}, C = {0, 12}
Salida: 6.5
Explicación:
El triángulo ABC forma un ángulo recto en el punto A. Por lo tanto, el ortocentro se encuentra en el punto A que es (0, 0).
La coordenada del circuncentro es (2.5, 6).
Por tanto, la distancia entre el ortocentro y el circuncentro es 6,5.Entrada: A = {0, 0}, B = {6, 0}, C = {0, 8}
Salida: 5
Explicación:
El triángulo ABC forma un ángulo recto en el punto A. Por lo tanto, el ortocentro se encuentra en el punto A que es (0, 0).
La coordenada del circuncentro es (3, 4).
Por lo tanto, la distancia entre el ortocentro y el circuncentro es 5.
Enfoque: La idea es encontrar las coordenadas del ortocentro y el circuncentro del triángulo dado en base a las siguientes observaciones:
El ortocentro es un punto donde se encuentran tres alturas. En un triángulo rectángulo, el ortocentro es el vértice que está situado en el vértice del ángulo recto .
El circuncentro es el punto donde se encuentra la mediatriz del triángulo. En un triángulo rectángulo, el circuncentro se encuentra en el centro de la hipotenusa .
Siga los pasos a continuación para resolver el problema:
- Encuentra el más largo de los tres lados del triángulo rectángulo, es decir, la hipotenusa.
- Encuentre el centro de la hipotenusa y póngalo como el circuncentro .
- Encuentra el vértice opuesto al lado más largo y configúralo como el ortocentro .
- Calcula la distancia entre ellos e imprímelo como resultado.
A continuación se muestra la implementación del enfoque anterior:
C++
// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
// Function to calculate Euclidean
// distance between the points p1 and p2
double distance(pair<double, double> p1,
pair<double, double> p2)
{
// Stores x coordinates of both points
double x1 = p1.first, x2 = p2.first;
// Stores y coordinates of both points
double y1 = p1.second, y2 = p2.second;
// Return the Euclid distance
// using distance formula
return sqrt(pow(x2 - x1, 2)
+ pow(y2 - y1, 2) * 1.0);
}
// Function to find orthocenter of
// the right angled triangle
pair<double, double>
find_orthocenter(pair<double, double> A,
pair<double, double> B,
pair<double, double> C)
{
// Find the length of the three sides
double AB = distance(A, B);
double BC = distance(B, C);
double CA = distance(C, A);
// Orthocenter will be the vertex
// opposite to the largest side
if (AB > BC && AB > CA)
return C;
if (BC > AB && BC > CA)
return A;
return B;
}
// Function to find the circumcenter
// of right angle triangle
pair<double, double>
find_circumcenter(pair<double, double> A,
pair<double, double> B,
pair<double, double> C)
{
// Circumcenter will be located
// at center of hypotenuse
double AB = distance(A, B);
double BC = distance(B, C);
double CA = distance(C, A);
// Find the hypotenuse and then
// find middle point of hypotenuse
// If AB is the hypotenuse
if (AB > BC && AB > CA)
return { (A.first + B.first) / 2,
(A.second + B.second) / 2 };
// If BC is the hypotenuse
if (BC > AB && BC > CA)
return { (B.first + C.first) / 2,
(B.second + C.second) / 2 };
// If AC is the hypotenuse
return { (C.first + A.first) / 2,
(C.second + A.second) / 2 };
}
// Function to find distance between
// orthocenter and circumcenter
void findDistance(pair<double, double> A,
pair<double, double> B,
pair<double, double> C)
{
// Find circumcenter
pair<double, double> circumcenter
= find_circumcenter(A, B, C);
// Find orthocenter
pair<double, double> orthocenter
= find_orthocenter(A, B, C);
// Find the distance between the
// orthocenter and circumcenter
double distance_between
= distance(circumcenter,
orthocenter);
// Print distance between orthocenter
// and circumcenter
cout << distance_between << endl;
}
// Driver Code
int main()
{
pair<double, double> A, B, C;
// Given coordinates A, B, and C
A = { 0.0, 0.0 };
B = { 6.0, 0.0 };
C = { 0.0, 8.0 };
// Function Call
findDistance(A, B, C);
return 0;
}
Java
// Java program for the above approach
import java.util.*;
class GFG{
static class pair
{
double first, second;
public pair(double first, double second)
{
this.first = first;
this.second = second;
}
}
// Function to calculate Euclidean
// distance between the points p1 and p2
static double distance(pair p1,
pair p2)
{
// Stores x coordinates of both points
double x1 = p1.first, x2 = p2.first;
// Stores y coordinates of both points
double y1 = p1.second, y2 = p2.second;
// Return the Euclid distance
// using distance formula
return Math.sqrt(Math.pow(x2 - x1, 2)
+ Math.pow(y2 - y1, 2) * 1.0);
}
// Function to find orthocenter of
// the right angled triangle
static pair
find_orthocenter(pair A,
pair B,
pair C)
{
// Find the length of the three sides
double AB = distance(A, B);
double BC = distance(B, C);
double CA = distance(C, A);
// Orthocenter will be the vertex
// opposite to the largest side
if (AB > BC && AB > CA)
return C;
if (BC > AB && BC > CA)
return A;
return B;
}
// Function to find the circumcenter
// of right angle triangle
static pair
find_circumcenter(pair A,
pair B,
pair C)
{
// Circumcenter will be located
// at center of hypotenuse
double AB = distance(A, B);
double BC = distance(B, C);
double CA = distance(C, A);
// Find the hypotenuse and then
// find middle point of hypotenuse
// If AB is the hypotenuse
if (AB > BC && AB > CA)
return new pair((A.first + B.first) / 2,
(A.second + B.second) / 2 );
// If BC is the hypotenuse
if (BC > AB && BC > CA)
return new pair((B.first + C.first) / 2,
(B.second + C.second) / 2 );
// If AC is the hypotenuse
return new pair( (C.first + A.first) / 2,
(C.second + A.second) / 2 );
}
// Function to find distance between
// orthocenter and circumcenter
static void findDistance(pair A,
pair B,
pair C)
{
// Find circumcenter
pair circumcenter
= find_circumcenter(A, B, C);
// Find orthocenter
pair orthocenter
= find_orthocenter(A, B, C);
// Find the distance between the
// orthocenter and circumcenter
double distance_between
= distance(circumcenter,
orthocenter);
// Print distance between orthocenter
// and circumcenter
System.out.print(distance_between +"\n");
}
// Driver Code
public static void main(String[] args)
{
pair A, B, C;
// Given coordinates A, B, and C
A = new pair( 0.0, 0.0 );
B = new pair(6.0, 0.0 );
C = new pair(0.0, 8.0 );
// Function Call
findDistance(A, B, C);
}
}
// This code is contributed by shikhasingrajput
Python3
# Python3 program for the above approach import math # Function to calculate Euclidean # distance between the points p1 and p2 def distance(p1, p2) : # Stores x coordinates of both points x1, x2 = p1[0], p2[0] # Stores y coordinates of both points y1, y2 = p1[1], p2[1] # Return the Euclid distance # using distance formula return int(math.sqrt(pow(x2 - x1, 2) + pow(y2 - y1, 2))) # Function to find orthocenter of # the right angled triangle def find_orthocenter(A, B, C) : # Find the length of the three sides AB = distance(A, B) BC = distance(B, C) CA = distance(C, A) # Orthocenter will be the vertex # opposite to the largest side if (AB > BC and AB > CA) : return C if (BC > AB and BC > CA) : return A return B # Function to find the circumcenter # of right angle triangle def find_circumcenter(A, B, C) : # Circumcenter will be located # at center of hypotenuse AB = distance(A, B) BC = distance(B, C) CA = distance(C, A) # Find the hypotenuse and then # find middle point of hypotenuse # If AB is the hypotenuse if (AB > BC and AB > CA) : return ((A[0] + B[0]) // 2, (A[1] + B[1]) // 2) # If BC is the hypotenuse if (BC > AB and BC > CA) : return ((B[0] + C[0]) // 2, (B[1] + C[1]) // 2) # If AC is the hypotenuse return ((C[0] + A[0]) // 2, (C[1] + A[1]) // 2) # Function to find distance between # orthocenter and circumcenter def findDistance(A, B, C) : # Find circumcenter circumcenter = find_circumcenter(A, B, C) # Find orthocenter orthocenter = find_orthocenter(A, B, C) # Find the distance between the # orthocenter and circumcenter distance_between = distance(circumcenter, orthocenter) # Print distance between orthocenter # and circumcenter print(distance_between) # Given coordinates A, B, and C A = [ 0, 0 ] B = [ 6, 0 ] C = [ 0, 8 ] # Function Call findDistance(A, B, C) # This code is contributed by divyesh072019.
C#
// C# program for the above approach
using System;
class GFG{
public class pair
{
public double first, second;
public pair(double first, double second)
{
this.first = first;
this.second = second;
}
}
// Function to calculate Euclidean
// distance between the points p1 and p2
static double distance(pair p1,
pair p2)
{
// Stores x coordinates of both points
double x1 = p1.first, x2 = p2.first;
// Stores y coordinates of both points
double y1 = p1.second, y2 = p2.second;
// Return the Euclid distance
// using distance formula
return Math.Sqrt(Math.Pow(x2 - x1, 2)
+ Math.Pow(y2 - y1, 2) * 1.0);
}
// Function to find orthocenter of
// the right angled triangle
static pair
find_orthocenter(pair A,
pair B,
pair C)
{
// Find the length of the three sides
double AB = distance(A, B);
double BC = distance(B, C);
double CA = distance(C, A);
// Orthocenter will be the vertex
// opposite to the largest side
if (AB > BC && AB > CA)
return C;
if (BC > AB && BC > CA)
return A;
return B;
}
// Function to find the circumcenter
// of right angle triangle
static pair
find_circumcenter(pair A,
pair B,
pair C)
{
// Circumcenter will be located
// at center of hypotenuse
double AB = distance(A, B);
double BC = distance(B, C);
double CA = distance(C, A);
// Find the hypotenuse and then
// find middle point of hypotenuse
// If AB is the hypotenuse
if (AB > BC && AB > CA)
return new pair((A.first + B.first) / 2,
(A.second + B.second) / 2 );
// If BC is the hypotenuse
if (BC > AB && BC > CA)
return new pair((B.first + C.first) / 2,
(B.second + C.second) / 2 );
// If AC is the hypotenuse
return new pair( (C.first + A.first) / 2,
(C.second + A.second) / 2 );
}
// Function to find distance between
// orthocenter and circumcenter
static void findDistance(pair A,
pair B,
pair C)
{
// Find circumcenter
pair circumcenter
= find_circumcenter(A, B, C);
// Find orthocenter
pair orthocenter
= find_orthocenter(A, B, C);
// Find the distance between the
// orthocenter and circumcenter
double distance_between
= distance(circumcenter,
orthocenter);
// Print distance between orthocenter
// and circumcenter
Console.Write(distance_between +"\n");
}
// Driver Code
public static void Main(String[] args)
{
pair A, B, C;
// Given coordinates A, B, and C
A = new pair( 0.0, 0.0 );
B = new pair(6.0, 0.0 );
C = new pair(0.0, 8.0 );
// Function Call
findDistance(A, B, C);
}
}
// This code is contributed by 29AjayKumar
Javascript
<script>
// Javascript implementation for the above approach
// Function to calculate Euclidean
// distance between the points p1 and p2
function distance( p1, p2)
{
// Stores x coordinates of both points
var x1 = p1[0], x2 = p2[0];
// Stores y coordinates of both points
var y1 = p1[1], y2 = p2[1];
// Return the Euclid distance
// using distance formula
return Math.sqrt(Math.pow(x2 - x1, 2)
+ Math.pow(y2 - y1, 2) * 1.0);
}
// Function to find orthocenter of
// the right angled triangle
function find_orthocenter( A, B, C)
{
// Find the length of the three sides
var AB = distance(A, B);
var BC = distance(B, C);
var CA = distance(C, A);
// Orthocenter will be the vertex
// opposite to the largest side
if (AB > BC && AB > CA)
return C;
if (BC > AB && BC > CA)
return A;
return B;
}
// Function to find the circumcenter
// of right angle triangle
function find_circumcenter( A, B, C)
{
// Circumcenter will be located
// at center of hypotenuse
var AB = distance(A, B);
var BC = distance(B, C);
var CA = distance(C, A);
// Find the hypotenuse and then
// find middle point of hypotenuse
// If AB is the hypotenuse
if (AB > BC && AB > CA)
return [ Math.floor((A[0] + B[0]) / 2),
Math.floor((A[1] + B[1]) / 2) ];
// If BC is the hypotenuse
if (BC > AB && BC > CA)
return [ Math.floor((B[0] + C[0]) / 2),
Math.floor((B[1] + C[1]) / 2) ];
// If AC is the hypotenuse
return [ Math.floor((C[0] + A[0]) / 2),
Math.floor((C[1] + A[1]) / 2) ];
}
// Function to find distance between
// orthocenter and circumcenter
function findDistance( A, B, C)
{
// Find circumcenter
circumcenter = find_circumcenter(A, B, C);
// Find orthocenter
orthocenter = find_orthocenter(A, B, C);
// Find the distance between the
// orthocenter and circumcenter
var distance_between
= distance(circumcenter,
orthocenter);
// Print distance between orthocenter
// and circumcenter
document.write(distance_between+"<br>");
}
// Driver Code
var A, B, C;
// Given coordinates A, B, and C
A = [ 0, 0 ];
B = [ 6, 0 ];
C = [ 0, 8 ];
// Function Call
findDistance(A, B, C);
// This code is contributed by Shubham Singh
</script>
Producción:
5
Tiempo Complejidad: O(1)
Espacio Auxiliar: O(1)
Publicación traducida automáticamente
Artículo escrito por AyushMalik y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA