MathDB

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

Source: KJMO 2008 p5

May 2, 2019

geometryTangentspentagonconcurrencyconcurrent

Problem Statement

Let there be a pentagon

ABCDE

inscribed in a circle

O

. The tangent to

O

E

is parallel to

AD

. A point

F

lies on

O

and it is in the opposite side of

A

with respect to

CD

, and satisfies

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

and

\angle CFD = 2\angle BFE

. Prove that the tangent to

O

B,E

and line

AF

concur at one point.