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National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2024 Sharygin Geometry Olympiad
21
21
Part of
2024 Sharygin Geometry Olympiad
Problems
(1)
I, Blaise Pascal, have a dream
Source: Sharygin Correspondence Round 2024 P21
3/6/2024
A chord
P
Q
PQ
PQ
of the circumcircle of a triangle
A
B
C
ABC
A
BC
meets the sides
B
C
,
A
C
BC, AC
BC
,
A
C
at points
A
′
,
B
′
A', B'
A
′
,
B
′
respectively. The tangents to the circumcircle at
A
A
A
and
B
B
B
meet at a point
X
X
X
, and the tangents at points
P
P
P
and
Q
Q
Q
meet at point
Y
Y
Y
. The line
X
Y
XY
X
Y
meets
A
B
AB
A
B
at a point
C
′
C'
C
′
. Prove that the lines
A
A
′
,
B
B
′
AA', BB'
A
A
′
,
B
B
′
and
C
C
′
CC'
C
C
′
concur.
geometry
circumcircle