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Classic problem #593MediumMathGeometry

Valid Square

Real-world version: Four Pegs Marking a Square Plot
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The statement here is our own: the same inputs, outputs and technique.

A land surveyor has hammered four pegs into a field and recorded each one as a coordinate pair [x, y] on the site grid. The pegs are handed to you as p1, p2, p3 and p4 in no particular order: the recording sequence says nothing about how the pegs sit relative to one another, so the corners of the plot may appear in any of the four positions.

Decide whether the four pegs are exactly the four corners of a square whose side length is greater than zero. All four sides must have the same length and all four corners must be right angles. Pegs that coincide, or that all lie on one straight line, cannot mark such a square.

Return true when the pegs mark a square plot and false otherwise.

Examples

Example 1

Input
p1 = [6, 1], p2 = [1, 6], p3 = [1, 1], p4 = [6, 6]
Output
true

Walking the pegs as `[1, 1]`, `[6, 1]`, `[6, 6]`, `[1, 6]` gives four sides of length 5 that meet at four right angles, so the pegs mark a square of side 5.

Example 2

Input
p1 = [2, 3], p2 = [2, 8], p3 = [2, 13], p4 = [9, 8]
Output
false

Three of the pegs sit on the vertical line `x = 2`, so no arrangement of the four gives a shape with four right angles.

Example 3

Input
p1 = [3, 0], p2 = [0, 3], p3 = [-3, 0], p4 = [0, -3]
Output
true

Walking the pegs as `[3, 0]`, `[0, 3]`, `[-3, 0]`, `[0, -3]` gives four sides that each span 3 across and 3 up or down, and each corner turns through a right angle. A square does not have to be lined up with the grid axes.

Constraints

  • p1.length == 2
  • p2.length == 2
  • p3.length == 2
  • p4.length == 2
  • -10^4 <= p1[i], p2[i], p3[i], p4[i] <= 10^4

The signature

The editor starts with this. It matches the parent problem's shape, so a solution that works here works on a standard judge unchanged.

Python
def is_square_plot(p1: list[int], p2: list[int], p3: list[int], p4: list[int]) -> bool:
Java
public boolean isSquarePlot(int[] p1, int[] p2, int[] p3, int[] p4)

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