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collinear wanted, inscribed ABCD and perpendiculars related

Source: Ukraine TST 2014 p8

May 1, 2020

geometrycollinearcyclic quadrilateralperpendicular

Problem Statement

The quadrilateral

ABCD

is inscribed in the circle

\omega

with the center

O

. Suppose that the angles

B

and

C

are obtuse and lines

AD

and

BC

are not parallel. Lines

AB

and

CD

intersect at point

E

. Let

P

and

R

be the feet of the perpendiculars from the point

E

on the lines

BC

and

AD

respectively.

Q

is the intersection point of

EP

and

AD, S

is the intersection point of

ER

and

BC

. Let K be the midpoint of the segment

QS

. Prove that the points

E, K

, and

O

are collinear.

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