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Saint Petersburg Mathematical Olympiad
2000 Saint Petersburg Mathematical Olympiad
9.2
9.2
Part of
2000 Saint Petersburg Mathematical Olympiad
Problems
(1)
Altitudes and midpoints. Prove that 4 points are concyclic.
Source: St. Petersburg MO 2000, 9th grade, P2
4/22/2023
Let
A
A
1
AA_1
A
A
1
and
C
C
1
CC_1
C
C
1
be altitudes of acute angled triangle
A
B
C
ABC
A
BC
. A point
D
D
D
is chosen on
A
A
1
AA_1
A
A
1
such that
A
1
D
=
C
1
D
A_1D=C_1D
A
1
D
=
C
1
D
. Let
E
E
E
be the midpoint of
A
C
AC
A
C
. Prove that points
A
A
A
,
C
1
C_1
C
1
,
D
D
D
,
E
E
E
are concylic.[I]Proposed by S. Berlov
geometry
Angle Chasing
St. Petersburg MO