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 1 of 7: Name the second intersections

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In plain words

The two new points let us transfer the perpendicular-bisector information to distances from A.

J=(FD\cap\Gamma)\setminus\{F\},\qquad K=(GE\cap\Gamma)\setminus\{G\}.

Detailed analysis

Let J be the point of Gamma different from F on line FD, and let K be the point of Gamma different from G on line GE. Thus F, D, J are collinear, G, E, K are collinear, and F, G, J, K all lie on Gamma.