MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece JBMO TST
2017 Greece JBMO TST
2
2
Part of
2017 Greece JBMO TST
Problems
(1)
show that quadrilateral is cyclic and its circumcircle contains the incenter
Source: Greece JBMO TST 2017, Problem 2
6/25/2018
Let
A
B
C
ABC
A
BC
be an acute-angled triangle inscribed in a circle
C
(
O
,
R
)
\mathcal C (O, R)
C
(
O
,
R
)
and
F
F
F
a point on the side
A
B
AB
A
B
such that
A
F
<
A
B
/
2
AF < AB/2
A
F
<
A
B
/2
. The circle
c
1
(
F
,
F
A
)
c_1(F, FA)
c
1
(
F
,
F
A
)
intersects the line
O
A
OA
O
A
at the point
A
′
A'
A
′
and the circle
C
\mathcal C
C
at
K
K
K
. Prove that the quadrilateral
B
K
F
A
′
BKFA'
B
K
F
A
′
is cyclic and its circumcircle contains point
O
O
O
.
geometry
Greece