Reasoning with circles

One idea does most of the work: radii are equal. Any triangle with two radii as sides is isosceles, with equal base angles. When a figure looks hard, join the centre to the points and look for those triangles. (When you are done, you can rub the helper lines out again.)

Angle in a semicircle (Fig. 5.30). Joining the centre O to A makes two isosceles triangles, with base angles a and b. The angle at A is a + b. The big triangle's angles give 2a + 2b = 180°, so a + b = 90°.

Opposite angles (Fig. 5.31). The radii make four isosceles triangles with base angles p, q, u and v. ∠A = p + v and ∠C = q + u. All four corners add to 2(p + q + u + v) = 360°, so ∠A + ∠C = 180°.

Which parallelogram fits in a circle? A parallelogram has equal opposite angles; a cyclic quadrilateral has opposite angles adding to 180°. Both at once means 90°: the only parallelogram that fits is a rectangle. Its diagonals subtend 90° at the circle, so both are diameters, crossing at the centre.

Longest and shortest chords:

• no chord is longer than a diameter: AB ≤ OA + OB = 2r;
• the shortest chord through a point A inside the circle is the one perpendicular to OA, because that chord is farthest from the centre.

A regular hexagon in a circle of radius r: each side subtends 60° at the centre, so each triangle is equilateral and each side is r, at distance (√3/2)r from the centre.