I am obsessed with this problem. 😪

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,𝑪 lie on a line in this order. 𝑨𝑩 is the diameter of semicircle 𝝎𝟏, 𝑨𝑪 is the
diameter of semicircle 𝝎𝟐. Assume both 𝝎𝟏 and 𝝎𝟐 are on the same side of 𝑨𝑪. 𝑫 is a point on 𝝎𝟐
such that 𝑩𝑫 ⊥ 𝑨𝑪. A circle centered at 𝑩 with radius 𝑩𝑫 intersects 𝝎𝟏 at 𝑬. 𝑭 is on 𝑨𝑪 such that
𝑬𝑭 ⊥ 𝑨𝑪. Prove that 𝑩𝑪 = 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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