Rectangle: equal diagonals

Depends on: SAS congruence, Parallelogram properties (opposite sides equal, diagonals bisect each other) (opposite sides equal). The definition of a rectangle (four right angles) is invoked directly from its primitive definition.

Statement

Let $ABCD$ be a rectangle, with vertices labelled in order ($A$, $B$, $C$, $D$ going around the rectangle, with adjacent labels at adjacent vertices). Its two diagonals are $AC$ and $BD$. Then the two diagonals have equal lengths:

By the definition of a rectangle, $ABCD$ is simultaneously a parallelogram (two pairs of opposite sides parallel) and has all four interior angles equal to $90^\circ$. These two facts are all the "extra information" used in the proof below.