MathDB

Circle and lines

Source: 2016 Taiwan TST 1st IMO Mock P1

July 8, 2016

geometry

Problem Statement

Let

AB

be a chord on a circle

O

,

M

be the midpoint of the smaller arc

AB

. From a point

C

outside the circle

O

draws two tangents to the circle

O

at the points

S

and

T

. Suppose

MS

intersects with

AB

at the point

E

,

MT

intersects with

AB

at the point

F

. From

E,F

draw a line perpendicular to

AB

that intersects with

OS,OT

at the points

X,Y

, respectively. Draw another line from

C

which intersects with the circle

O

at the points

P

and

Q

. Let

R

be the intersection point of

MP

and

AB

. Finally, let

Z

be the circumcenter of triangle

PQR

. Prove that

X

,

Y

and

Z

are collinear.

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