APMO · 7. Circles, cyclic quadrilaterals and tangents
Inscribed and tangent–chord angles with directed angles, hidden cyclic quadrilaterals, power of a point and radical axes, and tangent circles by homothety, through five complete APMO proofs.
Chase angles in circles with directed angles so that one proof covers every configuration.
Discover and prove hidden cyclic quadrilaterals and turn them into similar triangles and ratios.
Use the power of a point, radical axes and the radical centre for concurrence and fixed circles.
Prove that two circles are tangent with a homothety centred on a circle.
Write complete olympiad proofs that handle degenerate positions.