MathDB
Problems
Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad
1972 All Soviet Union Mathematical Olympiad
170
170
Part of
1972 All Soviet Union Mathematical Olympiad
Problems
(1)
ASU 170 All Soviet Union MO interior point isosceles triangles with all vertices
Source:
7/3/2019
The point
O
O
O
inside the convex polygon makes isosceles triangle with all the pairs of its vertices. Prove that
O
O
O
is the centre of the circumscribed circle.other formulation:
P
P
P
is a convex polygon and
X
X
X
is an interior point such that for every pair of vertices
A
,
B
A, B
A
,
B
, the triangle
X
A
B
XAB
X
A
B
is isosceles. Prove that all the vertices of
P
P
P
lie on a circle with center
X
X
X
.
isosceles
geometry