MathDB
Problems
Contests
International Contests
Kvant Problems
Kvant 2024
M2781
M2781
Part of
Kvant 2024
Problems
(1)
Circumcenter and incenter lie on an Euler line
Source: Kvant Magazine No. 1 2024 M2781
4/7/2024
Let
A
1
A_1
A
1
be the midpoint of the smaller arc
B
C
BC
BC
of the circumcircle of the acute-angled triangle
A
B
C
.
ABC.{}
A
BC
.
The point
A
1
A_1
A
1
is reflected relative to the side
B
C
,
BC,
BC
,
and then its image is reflected relative to the bisector of
∠
B
A
C
\angle BAC{}
∠
B
A
C
resulting in the point
A
2
A_2
A
2
. Similarly, the points
B
2
B_2
B
2
and
C
2
C_2
C
2
are constructed. Prove that the circumcenter and incenter of the triangle
A
B
C
ABC{}
A
BC
lie on the Euler line of the triangle
A
2
B
2
C
2
.
A_2B_2C_2.
A
2
B
2
C
2
.
Proposed by A. Tereshin
geometry
Euler Line