Cho đường tròn (O;R) đường kính AB cố định. Điểm I nằm giữa A và O. Dây CD vuông góc AB tại I. Gọi M là điểm tuỳ ý thuộc cung lớn CD (M không trùng với C,D và B). Dây AM cắt CD tại K

Để chứng minh các điều cần CM ta thực hiện các bước sau:

1) Ta có IKMB là tứ giác nội tiếp:
- Ta có $\angle{IKM} = \angle{AKM}$ (cùng chắn cung AM trên cùng đường tròn (O;R))
- Ta có $\angle{IMK} = \angle{AMK}$ (cùng chắn cung AK trên cùng đường tròn (O;R))
Vậy tứ giác IKMB nội tiếp.

- Ta có $AC^2 = AO^2 - OC^2 = R^2 - (R - R\cos{\angle{AOC}})^2 = R^2 - R^2(1 - \cos{\angle{AOC}})^2 = R^2 - R^2(1 - \frac{2S_{AOC}}{AO\cdot OC})^2 = R^2 - R^2(1 - \frac{2S_{AOC}}{2R\cdot R\sin{\angle{AOC}}})^2 = R^2 - R^2(1 - \frac{2S_{AOC}}{2R^2\sin{\angle{AOC}}})^2 = R^2 - R^2(1 - \frac{2S_{AOC}}{2R^2\cdot \frac{AC}{R}})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{R^2\cdot \frac{AC}{R}})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^2(1 - \frac{S_{AOC}}{AC\cdot R})^2 = R^2 - R^