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All-Russian Olympiad Regional Round
2000 All-Russian Olympiad Regional Round
11.6
11.6
Part of
2000 All-Russian Olympiad Regional Round
Problems
(1)
collinear, incircle and another circle - All-Russian MO 2000 Regional (R4) 11.6
Source:
9/26/2024
A circle inscribed in triangle
A
B
C
ABC
A
BC
has center
O
O
O
and touches side
A
C
AC
A
C
at point
K
K
K
. A second circle also has center
O
O
O
, intersects all sides of triangle
A
B
C
ABC
A
BC
. Let
E
E
E
and
F
F
F
be the corresponding points of intersection with sides
A
B
AB
A
B
and
B
C
BC
BC
, closest to vertex
B
B
B
;
B
1
B_1
B
1
and
B
2
B_2
B
2
are the points of its intersection with side
A
C
AC
A
C
, and
B
1
B_1
B
1
is closer to
A
A
A
. Prove that points
B
B
B
,
K
K
K
and point
P
P
P
, the intersections of the segments
B
2
E
B_2E
B
2
E
and
B
1
F
B_1F
B
1
F
lie on the same straight line.
geometry
collinear