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International Contests
International Zhautykov Olympiad
2023 International Zhautykov Olympiad
2
2
Part of
2023 International Zhautykov Olympiad
Problems
(1)
Isosceles triangle geo with parallel lines and tangencies
Source: IZHO 2023 P2
2/6/2023
The tangent at
C
C
C
to
Ω
\Omega
Ω
, the circumcircle of scalene triangle
A
B
C
ABC
A
BC
intersects
A
B
AB
A
B
at
D
D
D
. Through point
D
D
D
, a line is drawn that intersects segments
A
C
AC
A
C
and
B
C
BC
BC
at
K
K
K
and
L
L
L
respectively. On the segment
A
B
AB
A
B
points
M
M
M
and
N
N
N
are marked such that
A
C
∥
N
L
AC \parallel NL
A
C
∥
N
L
and
B
C
∥
K
M
BC \parallel KM
BC
∥
K
M
. Lines
N
L
NL
N
L
and
K
M
KM
K
M
intersect at point
P
P
P
lying inside the triangle
A
B
C
ABC
A
BC
. Let
ω
\omega
ω
be the circumcircle of
M
N
P
MNP
MNP
. Suppose
C
P
CP
CP
intersects
ω
\omega
ω
again at
Q
Q
Q
. Show that
D
Q
DQ
D
Q
is tangent to
ω
\omega
ω
.
geometry