Distance between orthocenter and circumcenter of a right-angled triangle
Given three pairs of integers A(x, y), B(x, y), and C(x, y), representing the coordinates of a right-angled triangle, the task is to find the distance between the orthocenter and circumcenter.
Examples:
Input: A = {0, 0}, B = {5, 0}, C = {0, 12}
Output: 6.5
Explanation:
Triangle ABC is right-angled at the point A. Therefore, orthocenter lies on the point A which is (0, 0).
The co-ordinate of circumcenter is (2.5, 6).
Therefore, the distance between the orthocenter and the circumcenter is 6.5.Input: A = {0, 0}, B = {6, 0}, C = {0, 8}
Output: 5
Explanation:
Triangle ABC is right-angled at the point A. Therefore, orthocenter lies on the point A which is (0, 0).
The co-ordinate of circumcenter is (3, 4).
Therefore, the distance between the orthocenter and the circumcenter is 5.
Approach: The idea is to find the coordinates of the orthocenter and the circumcenter of the given triangle based on the following observations:
The orthocenter is a point where three altitude meets. In a right angle triangle, the orthocenter is the vertex which is situated at the right-angled vertex.
The circumcenter is the point where the perpendicular bisector of the triangle meets. In a right-angled triangle, the circumcenter lies at the center of the hypotenuse.
Follow the steps below to solve the problem:
- Find the longest of the three sides of the right-angled triangle, i.e. the hypotenuse.
- Find the center of the hypotenuse and set it as the circumcenter.
- Find the vertex opposite to the longest side and set it as the orthocenter.
- Calculate the distance between them and print it as the result.
Below is the implementation of the above approach:
C++
// C++ program for the above approach#include <bits/stdc++.h>using namespace std;// Function to calculate Euclidean// distance between the points p1 and p2double 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 trianglepair<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 trianglepair<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 circumcentervoid 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 Codeint 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 approachimport 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 approachimport math# Function to calculate Euclidean# distance between the points p1 and p2def 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 triangledef 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 triangledef 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 circumcenterdef 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 CA = [ 0, 0 ]B = [ 6, 0 ]C = [ 0, 8 ]# Function CallfindDistance(A, B, C)# This code is contributed by divyesh072019. |
C#
// C# program for the above approachusing 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 p2function 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 trianglefunction 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 trianglefunction 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 circumcenterfunction 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 Codevar A, B, C; // Given coordinates A, B, and CA = [ 0, 0 ];B = [ 6, 0 ];C = [ 0, 8 ];// Function CallfindDistance(A, B, C);// This code is contributed by Shubham Singh</script> |
Output:
5
Time Complexity: O(1)
Auxiliary Space: O(1)
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