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butterfly theorem in finnish high school math competition

Source: Finland 2018, p3

September 8, 2019

geometryChordsmidpointcircle

Problem Statement

The chords

AB

and

CD

of a circle intersect at

M

, which is the midpoint of the chord

PQ

. The points

X

and

Y

are the intersections of the segments

AD

and

PQ

, respectively, and

BC

and

PQ

, respectively. Show that

M

is the midpoint of

XY