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Middle European Mathematical Olympiad
2022 Middle European Mathematical Olympiad
6
6
Part of
2022 Middle European Mathematical Olympiad
Problems
(1)
Tangent circles
Source: MEMO 2022 T6
9/2/2022
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral such that
A
C
=
B
D
AC = BD
A
C
=
B
D
and the sides
A
B
AB
A
B
and
C
D
CD
C
D
are not parallel. Let
P
P
P
be the intersection point of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
. Points
E
E
E
and
F
F
F
lie, respectively, on segments
B
P
BP
BP
and
A
P
AP
A
P
such that
P
C
=
P
E
PC=PE
PC
=
PE
and
P
D
=
P
F
PD=PF
P
D
=
PF
. Prove that the circumcircle of the triangle determined by the lines
A
B
,
C
D
,
E
F
AB, CD, EF
A
B
,
C
D
,
EF
is tangent to the circumcircle of the triangle
A
B
P
ABP
A
BP
.
geometry