MathDB
Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2021 Czech and Slovak Olympiad III A
6
6
Part of
2021 Czech and Slovak Olympiad III A
Problems
(1)
common interior point for all triangles by symmetrics of a point wrt sidelengths
Source: 2021 Czech and Slovak Olympiad III A p6
6/6/2021
An acute triangle
A
B
C
ABC
A
BC
is given. Let us denote
X
X
X
for each of its inner points
X
a
,
X
b
,
X
c
X_a, X_b, X_c
X
a
,
X
b
,
X
c
its images in axial symmetries sequentially along the lines
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
. Prove that all
X
a
X
b
X
c
X_aX_bX_c
X
a
X
b
X
c
triangles have a common interior point. (Josef Tkadlec)
geometry
Fixed point
fixed