Converse of the Pythagorean theorem

Depends on: Pythagorean theorem, SSS congruence, Perpendicular from a point to a line exists and is unique. Tacitly available in the proof: Ruler axiom (laying off a fixed length along a ray), Protractor axiom (the ray with a specified side and angle is unique).

Statement

Let the sides of $\triangle ABC$ be denoted $a = |BC|$, $b = |CA|$, $c = |AB|$. If

then

That is: the squared-edge relation alone already suffices to determine that one of the triangle's angles is a right angle.