The image contains a geometry problem involving a right-angled triangle ABC with various constructions and relationships. It asks for proofs related to points and lines in the triangle.

Here’s a summary of the problem:

1. Triangle ABC is a right triangle with a right angle at A.
2. Point M is chosen on the side AB such that AM = AC.
3. Line MK is parallel to AC and BC (with K on BC).
4. Line KI is perpendicular to AC at point I, and line ME intersects AB at point E.

The problem then asks for the following:

• (a) Proof that points A, I, H, K lie on a circle.
• (b) Proof that $MK^2 = ME \times MB$, and $CH \times CB = CK \times HA$.
• (c) Proof of the equation: $\frac{1}{AC^2} = \frac{1}{BC^2} + \frac{1}{KC^2}.$

Would you like me to walk through a specific part of the proof or explain any of the terms involved?