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 7 of 7: Conclude the required relation

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In plain words

The equal angles use the same line through E, G, and K as their reference, so the other sides have the same direction.

\angle KED=\angle KGF\quad\Longrightarrow\quad DE\parallel FG\text{ or }DE=FG.

Detailed analysis

The preceding steps give equality of the directed angles made by DE and FG with the common line E G K. Hence DE and FG are parallel. If the two lines coincide, this is the permitted second alternative in the statement.