I am obsessed with this problem. πŸ˜ͺ

It has been almost 10 months since I sat for bdmo 2023’s national. And couldn’t solve this geometry problem yet.Here is it-

Let the points 𝑨, 𝑩,π‘ͺ lie on a line in this order. 𝑨𝑩 is the diameter of semicircle 𝝎𝟏, 𝑨π‘ͺ is the
diameter of semicircle 𝝎𝟐. Assume both 𝝎𝟏 and 𝝎𝟐 are on the same side of 𝑨π‘ͺ. 𝑫 is a point on 𝝎𝟐
such that 𝑩𝑫 βŠ₯ 𝑨π‘ͺ. A circle centered at 𝑩 with radius 𝑩𝑫 intersects 𝝎𝟏 at 𝑬. 𝑭 is on 𝑨π‘ͺ such that
𝑬𝑭 βŠ₯ 𝑨π‘ͺ. Prove that 𝑩π‘ͺ = 𝑩𝑭.

Is there anyone to help?

Try Radical Axis Theorem.

Huh, I have proven it in an unimaginably easy way, :woozy_face:

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if you don’t mind,can you share with me?