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Romanian Masters of Mathematics Collection
2018 Romanian Master of Mathematics Shortlist
G1
G1
Part of
2018 Romanian Master of Mathematics Shortlist
Problems
(1)
Equal angles with a point on altitude
Source: RMM 2018 SL G1
2/21/2019
Let
A
B
C
ABC
A
BC
be a triangle and let
H
H
H
be the orthogonal projection of
A
A
A
on the line
B
C
BC
BC
. Let
K
K
K
be a point on the segment
A
H
AH
A
H
such that
A
H
=
3
K
H
AH = 3 KH
A
H
=
3
KH
. Let
O
O
O
be the circumcenter of triangle
A
B
C
ABC
A
BC
and let
M
M
M
and
N
N
N
be the midpoints of sides
A
C
AC
A
C
and
A
B
AB
A
B
respectively. The lines
K
O
KO
K
O
and
M
N
MN
MN
meet at a point
Z
Z
Z
and the perpendicular at
Z
Z
Z
to
O
K
OK
O
K
meets lines
A
B
,
A
C
AB, AC
A
B
,
A
C
at
X
X
X
and
Y
Y
Y
respectively. Show that
∠
X
K
Y
=
∠
C
K
B
\angle XKY = \angle CKB
∠
X
K
Y
=
∠
C
K
B
.Italy
geometry
circumcircle