File: valid-boomerang/solution.py

Date: 2026-06-06

Time: 19:37

Purpose

This file solves LeetCode 1037 — Valid Boomerang. It determines whether three 2D points form a "boomerang" — three distinct, non-collinear points. It owns exactly one responsibility: the collinearity check for a triple of points.

Key Components

Solution.isBoomerang

Contract: Takes a list of exactly three [x, y] coordinate pairs. Returns True if the three points are not collinear (i.e., they form a triangle with nonzero area), False otherwise.

The entire solution is a single expression — the cross product of two vectors:

(x2 - x1) * (y3 - y1) - (y2 - y1) * (x3 - x1) != 0

This computes the z-component of the cross product of vectors (P1→P2) and (P1→P3). If it's zero, the three points are collinear (or at least two are identical); if nonzero, they form a valid boomerang.

Patterns

Cross-product collinearity test — instead of computing slopes (which requires division and special-casing vertical lines / division by zero), the solution uses the cross product. This is the standard idiom for collinearity in computational geometry: it's branchless, works for all orientations including vertical/horizontal lines, and uses only integer arithmetic when inputs are integers, avoiding floating-point precision issues entirely.

Destructuring assignment — the tuple unpack (x1, y1), (x2, y2), (x3, y3) = points gives each coordinate a readable name in one line.

Dependencies

Imports: typing.List — used only for the type annotation on the method signature.

Imported by: valid-boomerang/test_solution.py directly. The "Imported By" list in the prompt is misleading — it shows every test file across the entire repo because they all import Solution from their respective solution.py, not from this file specifically.

Flow

1. Destructure the three points into six scalar coordinates.

2. Compute the cross product of vectors P1→P2 and P1→P3.

3. Return whether the result is nonzero.

No loops, no branches, no allocations. O(1) time and space.

Invariants

• The cross product is zero if and only if the points are collinear. This covers all degenerate cases: two identical points, three identical points, and three distinct but collinear points.
• Integer arithmetic only (assuming integer inputs from LeetCode constraints), so there are no floating-point rounding concerns.

Error Handling

None. The function assumes exactly three points, each with exactly two coordinates, as guaranteed by the LeetCode problem constraints. A malformed input (fewer points, wrong dimensions) would raise a ValueError from the destructuring unpack — this is fine since the caller is responsible for honoring the contract.

Topics to Explore

• [file] valid-boomerang/test_solution.py — See what edge cases are covered (duplicate points, horizontal/vertical lines, etc.)
• [file] check-if-it-is-a-straight-line/solution.py — Uses the same cross-product collinearity test generalized to N points
• [file] largest-triangle-area/solution.py — Uses the cross product to compute area rather than just checking for zero
• [general] cross-product-collinearity — Why cross product beats slope comparison: no division-by-zero edge cases, exact integer arithmetic, single expression

Beliefs

• boomerang-cross-product-equivalence — The cross product being nonzero is equivalent to the three points being distinct and non-collinear; no separate duplicate-point check is needed.
• boomerang-integer-only — The solution uses only integer multiplication and subtraction, avoiding all floating-point precision issues.
• boomerang-constant-complexity — isBoomerang executes in O(1) time and O(1) space with no loops or conditionals.
• boomerang-no-slope-division — The solution avoids slope-based comparison (dy/dx), which would require handling vertical lines and division by zero as special cases.