MathDB
Problems
Contests
International Contests
Balkan MO Shortlist
2022 Balkan MO Shortlist
G4
G4
Part of
2022 Balkan MO Shortlist
Problems
(1)
Tangency geo
Source: BMO Shortlist 2022, G4
5/13/2023
Let
A
B
C
ABC
A
BC
be a triangle and let the tangent at
B
B{}
B
to its circumcircle meet the internal bisector of the angle
A
A{}
A
at
P
P{}
P
. The line through
P
P{}
P
parallel to
A
C
AC
A
C
meets
A
B
AB
A
B
at
Q
Q{}
Q
. Assume that
Q
Q{}
Q
lies in the interior of segment
A
B
AB
A
B
and let the line through
Q
Q{}
Q
parallel to
B
C
BC
BC
meet
A
C
AC
A
C
at
X
X{}
X
and
P
C
PC
PC
at
Y
Y{}
Y
. Prove that
P
X
PX
PX
is tangent to the circumcircle of the triangle
X
Y
C
XYC
X
Y
C
.
geometry