Square props — Square: union of rectangle ∩ rhombus · Principia

Dependencies: Rectangle: equal diagonals (Rectangle: equal diagonals), Rhombus: diagonals ⊥-bisect (Rhombus: diagonals ⊥-bisect), and the prerequisites they each cite ("a rectangle has 4 right angles" and "a rhombus has 4 equal sides").

Statement

Let $ABCD$ be a square — by definition, a quadrilateral that is both a rectangle and a rhombus. Then $ABCD$ inherits all the properties of both rectangles and rhombi, written together as:

\begin{aligned} &\text{(R1)}\ \angle A = \angle B = \angle C = \angle D = 90^{\circ},\\ &\text{(R2)}\ |AC| = |BD|,\\ &\text{(M1)}\ |AB| = |BC| = |CD| = |DA|,\\ &\text{(M2)}\ AC \perp BD,\quad AC,\,BD\ \text{bisect each other},\\ &\text{(M3)}\ AC\ \text{bisects}\ \angle BAD\ \text{and}\ \angle BCD,\quad BD\ \text{bisects}\ \angle ABC\ \text{and}\ \angle ADC. \end{aligned}

(R) comes from the rectangle side; (M) comes from the rhombus side; the square directly merges the two lists.

A square ABCD wears both labels at once: "rectangle (red = 4 right angles + equal diagonals)" and "rhombus (blue = 4 equal sides + ⊥ + 45° half-angles)".

Square properties (= rectangle ∩ rhombus ⇒ union of properties)

Statement

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