5

Part of 2008 Korea Junior Math Olympiad

Problems(1)

two tangents of the same circle and another line are concurrent, equal products

Source: KJMO 2008 p5

Let there be a pentagon

ABCDE

inscribed in a circle

O

. The tangent to

O

E

AD

F

O

and it is in the opposite side of

\angle CFD = 2\angle BFE

with respect to

CD

, and satisfies

AB \cdot BC \cdot DF = AE \cdot ED \cdot CF

\angle CFD = 2\angle BFE

. Prove that the tangent to

O

B,E

AF

concur at one point.

geometryTangentspentagonconcurrencyconcurrent