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 4 of 7: Find a circle centered at A

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In plain words

All four points D, E, J, K are now at the same distance from A.

AD=AE=AK=AJ.

Detailed analysis

The hypothesis gives AD equal to AE. The preceding two steps give AJ equal to AD and AK equal to AE. Consequently D, E, J, K lie on one circle whose center is A.