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isosceles and ratio wanted, Puerto Rico TST 2022.2.3

Source:

March 23, 2024
ratiogeometryisosceles

Problem Statement

Let ω\omega be a circle with center OO and diameter ABAB. A circle with center at BB intersects ω\omega at C and ABAB at DD. The line CDCD intersects ω\omega at a point EE (E≠CE\ne C). The intersection of lines OEOE and BCBC is FF. (a) Prove that triangle OBFOBF is isosceles. (b) If DD is the midpoint of OBOB, find the value of the ratio FBBD\frac{FB}{BD}.
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