MathDB
Problems
Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2022 Sharygin Geometry Olympiad
9.7
9.7
Part of
2022 Sharygin Geometry Olympiad
Problems
(1)
KR is perpendicular to a median in ABC
Source: Sharygin Finals 2022 9.7
8/1/2022
Let
H
H
H
be the orthocenter of an acute-angled triangle
A
B
C
ABC
A
BC
. The circumcircle of triangle
A
H
C
AHC
A
H
C
meets segments
A
B
AB
A
B
and
B
C
BC
BC
at points
P
P
P
and
Q
Q
Q
. Lines
P
Q
PQ
PQ
and
A
C
AC
A
C
meet at point
R
R
R
. A point
K
K
K
lies on the line
P
H
PH
P
H
in such a way that
∠
K
A
C
=
9
0
∘
\angle KAC = 90^{\circ}
∠
K
A
C
=
9
0
∘
. Prove that
K
R
KR
K
R
is perpendicular to one of the medians of triangle
A
B
C
ABC
A
BC
.
geometry