Equal chords, equal angles

Tie a thread tightly between two points on a wheel: it is a chord. Turn the wheel. The thread is now a different chord of the same length, and it seems to make the same angle at the centre.

Every chord makes an isosceles triangle with the centre, because two of its sides are radii. That one fact drives this whole lesson.

Theorem 2: equal chords subtend equal angles at the centre. Take chords AB = DE in a circle with centre C:

• CA = CB = CD = CE = r (all radii);
• AB = DE (given);
• so ΔCAB ≅ ΔCDE by SSS, and ∠ACB = ∠DCE.

Theorem 3 is the converse: chords that subtend equal angles at the centre are equal. If ∠ACB = ∠DCE, then CA = CD, CB = CE and the angle between them is equal, so ΔACB ≅ ΔDCE by SAS, and AB = DE.

Statement and converse: swap the two boxes. Theorem 2 says if [equal chords] then [equal angles]. Theorem 3 swaps them: if [equal angles] then [equal chords]. A converse needs its own proof; here it uses SAS instead of SSS.

Picture standing at the centre with your arms held at a fixed angle while the floor turns under you. Your hands trace chords of the same length.