It looks like you've uploaded a math problem involving geometry. I'll break down the tasks based on the content of the image:

1. Prove that triangles ADE and CDA are similar.
2. Prove that $BD \times BC = BE \times CD$.
3. Prove that $HEF$ is a right triangle and show that $EF$ is the bisector of angle HEB.
4. Use the Pythagorean theorem to show that: $\frac{1}{BF} + \frac{1}{BC} = \frac{1}{BK}$

Let's start with the first problem: Proving that triangles ADE and CDA are similar.

For that, we can explore their corresponding angles and sides. I'll proceed step by step.

Would you like me to begin with that? Let me know how you would like to proceed or if you'd like me to go over any of the details from the problem first.