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Three of four points on a circle determine a 108° angle

Source: Czech-Polish-Slovak Match, 2011

August 9, 2011

geometrygeometric transformationreflectionperpendicular bisectorgeometry unsolved

Problem Statement

Points

A

,

B

,

C

,

D

lie on a circle (in that order) where

AB

and

CD

are not parallel. The length of arc

AB

(which contains the points

D

and

C

) is twice the length of arc

CD

(which does not contain the points

A

and

B

). Let

E

be a point where

AC=AE

and

BD=BE

. Prove that if the perpendicular line from point

E

to the line

AB

passes through the center of the arc

CD

(which does not contain the points

A

and

B

), then

\angle ACB = 108^\circ

.

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