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National and Regional Contests
Russia Contests
Russian Team Selection Tests
Russian TST 2019
P3
P3
Part of
Russian TST 2019
Problems
(1)
Tangent circumcircles
Source: Russian TST 2019, Day 9 P3 (Groups A & B)
3/27/2023
Let
H
H{}
H
be the orthocenter of the acute-angled triangle
A
B
C
ABC
A
BC
. In the triangle
B
H
C
BHC
B
H
C
, the median
H
M
HM
H
M
and the symedian
H
L
HL
H
L
are drawn. The point
K
K{}
K
is marked on the line
L
H
LH
L
H
so that
∠
A
K
L
=
9
0
∘
\angle AKL=90^\circ
∠
A
K
L
=
9
0
∘
. Prove that the circumcircles of the triangles
A
B
C
ABC
A
BC
and
K
L
M
KLM
K
L
M
are tangent.
geometry