Valid Square

Valid Square - Problem

Math Geometry Medium

Given the coordinates of four points in 2D space p1, p2, p3 and p4, return true if the four points construct a square.

The coordinate of a point p_i is represented as [x_i, y_i]. The input is not given in any order.

A valid square has four equal sides with positive length and four equal angles (90-degree angles).

Input & Output

Example 1 — Valid Square

$ Input: p1 = [0,0], p2 = [1,1], p3 = [1,0], p4 = [0,1]

› Output: true

💡 Note: The four points form a unit square. All sides have length 1, and both diagonals have length √2.

Example 2 — Invalid Square

$ Input: p1 = [0,0], p2 = [1,1], p3 = [1,0], p4 = [0,12]

› Output: false

💡 Note: The points do not form a square because the sides have different lengths.

Example 3 — Collinear Points

$ Input: p1 = [1,0], p2 = [2,0], p3 = [3,0], p4 = [4,0]

› Output: false

💡 Note: All four points lie on the same line, so they cannot form a square.

Constraints

The coordinates of all four points are integers
-10⁴ ≤ x_i, y_i ≤ 10⁴
All four points are distinct

Visualization

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Asked in

G Google 25 M Microsoft 20 A Apple 15 F Facebook 12

The key insight is that a valid square has exactly 4 equal sides and 2 equal diagonals, with diagonal length = √2 × side length. Best approach is to calculate all 6 distances between points and verify these geometric properties. Time: O(1), Space: O(1)

Common Approaches

✓ Center and Side Length Check

⏱️ Time: O(1) Space: O(1)

Calculate the center of all four points and check if all points are at the same distance from center. Then verify the angle relationships for a square.

Distance Calculation

⏱️ Time: O(1) Space: O(1)

Calculate distances between all pairs of points. A valid square should have 4 equal sides and 2 equal diagonals, where diagonal = side × √2.

Center and Side Length Check — Algorithm Steps

Find the center point of all four points
Calculate distance from center to each point
Check if all distances are equal
Verify that adjacent sides are perpendicular

Visualization

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Step-by-Step Walkthrough

Calculate All Distances

Get 6 distances between all point pairs

Group by Sets

Use sets to check if sides and diagonals are uniform

Validate Ratio

Ensure diagonal = √2 × side length

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

long long distanceSquared(int* p1, int* p2) {
    long long dx = p1[0] - p2[0];
    long long dy = p1[1] - p2[1];
    return dx * dx + dy * dy;
}

int compare(const void* a, const void* b) {
    long long la = *(long long*)a;
    long long lb = *(long long*)b;
    if (la < lb) return -1;
    if (la > lb) return 1;
    return 0;
}

int solution(int* p1, int* p2, int* p3, int* p4) {
    int* points[4] = {p1, p2, p3, p4};
    long long distances[6];
    int idx = 0;
    
    // Calculate all 6 distances
    for (int i = 0; i < 4; i++) {
        for (int j = i + 1; j < 4; j++) {
            distances[idx++] = distanceSquared(points[i], points[j]);
        }
    }
    
    qsort(distances, 6, sizeof(long long), compare);
    
    // Check if first 4 distances are equal (sides) and last 2 are equal (diagonals)
    return (distances[0] == distances[1] && distances[1] == distances[2] && 
            distances[2] == distances[3] && distances[4] == distances[5] &&
            distances[0] > 0 && distances[4] == 2 * distances[0]);
}

void parsePoint(char* line, int* point) {
    char* token = strtok(line, "[,]");
    point[0] = atoi(token);
    token = strtok(NULL, "[,]");
    point[1] = atoi(token);
}

int main() {
    char line[100];
    int p1[2], p2[2], p3[2], p4[2];
    
    fgets(line, sizeof(line), stdin);
    parsePoint(line, p1);
    fgets(line, sizeof(line), stdin);
    parsePoint(line, p2);
    fgets(line, sizeof(line), stdin);
    parsePoint(line, p3);
    fgets(line, sizeof(line), stdin);
    parsePoint(line, p4);
    
    int result = solution(p1, p2, p3, p4);
    printf(result ? "true\n" : "false\n");
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(1)

Fixed 4 points means constant number of operations regardless of input size

✓ Linear Growth

Space Complexity

O(1)

Only using variables for center coordinates and distances, no extra data structures

✓ Linear Space

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Input & Output

Constraints

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Related Problems

Common Approaches

Center and Side Length Check — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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