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Problems
Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2024 Sharygin Geometry Olympiad
14
14
Part of
2024 Sharygin Geometry Olympiad
Problems
(1)
Feuerbach point in a right angled triangle.
Source: Sharygin Correspondence Round 2024 P14
3/6/2024
The incircle
ω
\omega
ω
of triangle
A
B
C
ABC
A
BC
, right angled at
C
C
C
, touches the circumcircle of its medial triangle at point
F
F
F
. Let
O
E
OE
OE
be the tangent to
ω
\omega
ω
from the midpoint
O
O
O
of the hypotenuse
A
B
AB
A
B
, distinct from
A
B
AB
A
B
. Prove that
C
E
=
C
F
CE = CF
CE
=
CF
.
geometry