Congruence
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
Euclid proves it without moving anything
The rearrangement proof cuts and slides. Euclid's does neither — it shows that a square and a rectangle are each exactly twice the same triangle, seen from opposite sides, and that is harder to hold in the head for a reason worth understanding.
The rope that squares a corner
The theorem turns two sides into a third. Run it backwards and it turns three lengths into a right angle — which is a different statement, needs its own proof, and is the only one of the two that has ever been used to build anything.
Named alongside it
The objects these essays reach for when they reach for this one.
AltitudeAreaConstructionConverseDissectionHypotenuseLaw of cosinesParallel linesPythagorean theoremPythagorean triplesRearrangementRight angle