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 2 of 7: Obtain the first equal distance

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In plain words

The perpendicular bisector makes triangle BFD isosceles, while the circumcircle supplies the matching angles.

FB=FD,\qquad \triangle BFD\sim\triangle JAD,\qquad JA=AD.

Detailed analysis

Because F lies on the perpendicular bisector of BD, FB equals FD. Since A, B, F, J are concyclic and A, D, B are collinear, the angle at F of triangle BFD equals the angle at A of triangle JAD. Reversing both rays along the two collinear lines also makes the angle at D equal. Hence the two triangles are similar, with FB corresponding to JA and FD corresponding to AD. The equality FB equals FD therefore gives JA equals AD.