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KoMaL A Problems
KoMaL A Problems 2018/2019
A. 742
A. 742
Part of
KoMaL A Problems 2018/2019
Problems
(1)
Collinearity in a twist on cyclic quad + incenters diagram
Source: KöMaL A. 742
2/13/2019
Convex quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in circle
Ω
\Omega
Ω
. Its sides
A
D
AD
A
D
and
B
C
BC
BC
intersect at point
E
E
E
. Let
M
M
M
and
N
N
N
be the midpoints of the circle arcs
A
B
AB
A
B
and
C
D
CD
C
D
not containing the other vertices, and let
I
I
I
,
J
J
J
,
K
K
K
,
L
L
L
denote the incenters of triangles
A
B
D
ABD
A
B
D
,
A
B
C
ABC
A
BC
,
B
C
D
BCD
BC
D
,
C
D
A
CDA
C
D
A
, respectively. Suppose
Ω
\Omega
Ω
intersects circles
I
J
M
IJM
I
J
M
and
K
L
N
KLN
K
L
N
for the second time at points
U
≠
M
U \neq M
U
=
M
and
V
≠
N
V \neq N
V
=
N
. Show that the points
E
E
E
,
U
U
U
, and
V
V
V
are collinear.
geometry