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 6 of 7: Transfer the angle to the original circumcircle

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In plain words

Collinearity changes the first angle, and concyclicity changes the second one.

\angle KJD=\angle KJF=\angle KGF.

Detailed analysis

Since F, D, J are collinear, the directed angle KJD equals KJF. Since F, G, J, K all lie on Gamma, the angles KJF and KGF subtend the same chord KF of Gamma and are equal. Combining the two equalities gives the displayed chain.