Valid Boomerang - Video Solutions

Problem Overview: You are given three points on a 2D plane. A boomerang is valid only if the points are all distinct and they do not lie on the same straight line. The task is to determine whether these three coordinates form a non‑collinear triangle.

The core observation: three points form a straight line if their slopes match or if the area of the triangle formed by them equals zero. Both approaches rely on basic geometry and simple math operations on coordinates stored in an array.

Approach 1: Using Slope to Determine Collinearity (Time: O(1), Space: O(1))

Two line segments share the same slope when three points are collinear. Compute the slope between point A→B and A→C and compare them. Instead of dividing (which risks floating‑point errors or division by zero), use cross multiplication: (y2 - y1) * (x3 - x1) == (y3 - y1) * (x2 - x1). If this equality holds, the slopes are identical and the points lie on the same line. Otherwise, they form a valid boomerang.

This method relies purely on integer arithmetic, which avoids precision problems and keeps the implementation clean. Since the input always contains exactly three points, you perform only a constant number of operations. That makes the runtime O(1) and space O(1). This is usually the most straightforward way to solve the problem.

Approach 2: Using Area of Triangle Formed by Points for Collinearity (Time: O(1), Space: O(1))

Three points are collinear when the triangle they form has zero area. Using the determinant formula for the area of a triangle with coordinates:

Area = x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)

If the computed value equals zero, the points lie on a straight line and the boomerang is invalid. Any non‑zero value means the points form a triangle, which satisfies the boomerang condition. This technique is mathematically equivalent to the slope method but framed through geometric area.

The computation uses only a few arithmetic operations and no additional memory, giving the same O(1) time and O(1) space complexity. Some engineers prefer this approach because the determinant formula is widely used in computational geometry.

Recommended for interviews: The slope comparison using cross multiplication is typically expected. It directly expresses the definition of collinearity and avoids floating‑point division. The triangle area formula demonstrates deeper geometry intuition, but both solutions run in constant time. Showing either approach signals strong fundamentals with coordinate geometry and basic mathematical reasoning.