Problem 1

Let

\Gamma

be the circumcircle of acute triangle

ABC

. Points

D

and

E

are on segments

AB

and

AC

, respectively, such that

AD=AE

. The perpendicular bisectors of

BD

and

CE

intersect the minor arcs

AB

and

AC

\Gamma

at points

F

and

G

, respectively. Prove that lines

DE

and

FG

are either parallel or they are the same line.

Step 5 of 7: Use the circle centered at A

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In plain words

Equal inscribed angles subtend the same chord in the new circle.

\angle KED=\angle KJD.

Detailed analysis

In the circle through D, E, J, K, the angles KED and KJD stand on the same chord KD. We use directed angles modulo 180 degrees, so the equality is valid regardless of which side of the chord the vertices occupy.