Square tests

Dependencies: Parallelogram tests, Rectangle tests, Rhombus tests.

Statement

Let $ABCD$ be a quadrilateral (vertices in cyclic order). The following three conditions are mutually equivalent, and each can serve as a test for "$ABCD$ is a square":

(a) A rectangle with a pair of equal adjacent sides: $ABCD$ is a rectangle and $|AB| = |BC|$; (b) A rhombus with one right angle: $ABCD$ is a rhombus and $\angle ABC = 90^\circ$; (c) Diagonals equal and perpendicular bisectors of each other: let the diagonals $AC$, $BD$ meet at $M$; then $|AM| = |MC|$, $|BM| = |MD|$, $|AC| = |BD|$, and $AC \perp BD$.

A more intuitive way to put it: square = rectangle ∩ rhombus. The essence of every test is to first use the four parallelogram tests to lock $ABCD$ as a parallelogram, then layer on the "special angle" or "special side length" condition required by the rectangle and rhombus sides respectively; the double upgrade yields a square.