MathDB

2 lines and a circle concurrent, 2 circumcircles and parallelogram related

Source: Dutch 2019 IMO TST2 p3

August 2, 2019

geometryconcurrencyconcurrentparallelogramcircumcircle

Problem Statement

Let

ABC

be an acute angles triangle with

O

the center of the circumscribed circle. Point

Q

lies on the circumscribed circle of

\vartriangle BOC

so that

OQ

is a diameter. Point

M

lies on

CQ

and point

N

lies internally on line segment

BC

so that

ANCM

is a parallelogram. Prove that the circumscribed circle of

\vartriangle BOC

and the lines

AQ

and

NM

pass through the same point.