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Kvant 2023
M2732
M2732
Part of
Kvant 2023
Problems
(1)
Circumcenter becomes orthocenter
Source: Kvant Magazine No. 1 2023 M2732
3/7/2023
Let
O
O{}
O
be the circumcenter of the triangle
A
B
C
ABC
A
BC
. On the rays
A
C
AC
A
C
and
B
C
BC
BC
consider the points
C
a
C_a
C
a
and
C
b
C_b
C
b
respectively, such that
A
C
a
AC_a
A
C
a
and
B
C
b
BC_b
B
C
b
are equal in length to
A
B
AB
A
B
. Let
O
c
O_c{}
O
c
be the circumcenter of the triangle
C
C
a
C
b
CC_aC_b
C
C
a
C
b
. Define the points
O
a
O_a{}
O
a
and
O
b
O_b{}
O
b
similarly. Prove that
O
O{}
O
is the orthocenter of the triangle
O
a
O
b
O
c
O_aO_bO_c
O
a
O
b
O
c
.Proposed by A. Zaslavsky
Kvant
geometry