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angle bisector wanted, tangents to semicircle (O)

Source: Indonesia INAMO Shortlist 2017 G7 https://artofproblemsolving.com/community/c1101409_indonesia_shortlist__geometry

November 15, 2021

geometryangle bisector

Problem Statement

A semicircle

(O)

is drawn with the center

O

, where

O

lies on a line

\ell

.

C

and

D

lie on the circle

(O)

, and the tangent lines of

(O)

at points

C

and

D

intersects the line

BD

at points

B

and

A

, respectively, such that

O

lies between points

B

and

A

. Let

E

be the intersection point between

AC

and

BD

, and

F

the point on

\ell

so that

EF

is perpendicular to line

\ell

. Prove that

EF

bisects the angle

\angle CFD

.

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