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Turkey Contests
Turkey Team Selection Test
2016 Turkey Team Selection Test
1
1
Part of
2016 Turkey Team Selection Test
Problems
(1)
Property of Orthocenter
Source: Turkey TST 2016 P1
4/10/2016
In an acute triangle
A
B
C
ABC
A
BC
, a point
P
P
P
is taken on the
A
A
A
-altitude. Lines
B
P
BP
BP
and
C
P
CP
CP
intersect the sides
A
C
AC
A
C
and
A
B
AB
A
B
at points
D
D
D
and
E
E
E
, respectively. Tangents drawn from points
D
D
D
and
E
E
E
to the circumcircle of triangle
B
P
C
BPC
BPC
are tangent to it at points
K
K
K
and
L
L
L
, respectively, which are in the interior of triangle
A
B
C
ABC
A
BC
. Line
K
D
KD
KD
intersects the circumcircle of triangle
A
K
C
AKC
A
K
C
at point
M
M
M
for the second time, and line
L
E
LE
L
E
intersects the circumcircle of triangle
A
L
B
ALB
A
L
B
at point
N
N
N
for the second time. Prove that
K
D
M
D
=
L
E
N
E
⟺
Point P is the orthocenter of triangle ABC
\frac{KD}{MD}=\frac{LE}{NE} \iff \text{Point P is the orthocenter of triangle ABC}
M
D
KD
=
NE
L
E
⟺
Point P is the orthocenter of triangle ABC
geometry