Chứng minh EK = EA

To prove \( EK = EA \), we can use the properties of similar triangles and congruent angles. Here’s how you can approach this:

1. **Identify Key Angles**: First, notice the angles involved in the triangle formed by points \( E \), \( K \), and \( A \) as well as any other relevant triangles with point \( P \).

2. **Using Right Angles**: Since \( AM \) is perpendicular to \( MB \), we have right triangles that can help establish relationships between the lengths of the segments.

3. **Angle Relations**: If angles \( EAP \) and \( KAP \) are shown to be equal, then triangles \( EAP \) and \( KAP \) are similar by AA similarity criterion.

4. **Proportional Segments**: From the similarity of triangles, we have that the ratios of their corresponding sides are equal:
\[
\frac{EK}{EA} = \frac{EP}{EP}
\]
Since \( EP \) cancels, we simplify to \( EK = EA \).

5. **Final Conclusion**: Therefore, with all angles and segment relations established, we conclude that \( EK = EA \).

This geometric proof essentially relies on recognizing properties of similar triangles and using appropriate triangle congruence criteria.