Geometry Dilemma.

Geometry Level 3

Let points B B and C C lie on the circle with diameter A D AD and center O O on the same side of A D AD . The circumcircles of triangles A B O ABO and C D O CDO meet B C BC at points F F and E E respectively.

Given A F = 49 AF = 49 and D E = 16 DE = 16 , find the radius of the largest circle with diameter A D AD .


The answer is 28.

Priyanshu Mishra by Priyanshu Mishra , 20, India

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1 solution

Δ O E D ∼ Δ F O A ⇒ A O × D O = A F × D E ⇒ A O 2 = 49 × 16 ⇒ O A O = 28 \Delta OED \sim \Delta FOA \Rightarrow AO \times DO = AF \times DE \Rightarrow AO^{2} = 49 \times 16 \Rightarrow OAO = 28 .

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