Why the Sides Match

Part of the Transformations & Constructions section of Coddy's Geometry journey. Lesson 37 of 60.

Why is the constructed triangle equilateral? Because of one rule: Radii are equal. Every radius of a circle, a segment from its center to the circle, has the same length.

$Triangle PQR inside the circles about P and about Q$

In triangle PQR, the sides PQ and PR both run from P to the circle about P, so PR = PQ. The sides QP and QR both run from Q to the circle about Q, so QR = QP. All three sides equal PQ. This is the first construction in Euclid's Elements, written about 2,300 years ago.

The construction is only trusted because each step has a reason, just like an angle chase.

Challenge

C was built from two circles, one about A through B and one about B through A. The sides BC and BA glow. Pick the rule that makes them equal, then press Check.

Try it yourself

$Name the rule that links them$