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lines intersect on circle, isosceles trapezoid, incircles related

Source: 2013 Oral Moscow Geometry Olympiad grades 10-11 p6

August 17, 2019

geometrytrapezoidincircleconcurrencyconcurrentarc midpoint

Problem Statement

The trapezoid

ABCD

is inscribed in the circle

w

(

AD // BC

). The circles inscribed in the triangles

ABC

and

ABD

touch the base of the trapezoid

BC

and

AD

at points

P

and

Q

respectively. Points

X

and

Y

are the midpoints of the arcs

BC

and

AD

of circle

w

that do not contain points

A

and

B

respectively. Prove that lines

XP

and

YQ

intersect on the circle

w