MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2018 China National Olympiad
4
4
Part of
2018 China National Olympiad
Problems
(1)
Incenters and concyclic points
Source: China Mathematical Olympiad 2018 Q4
11/16/2017
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral whose diagonals intersect at
P
P
P
. The circumcircle of
△
A
P
D
\triangle APD
△
A
P
D
meets segment
A
B
AB
A
B
at points
A
A
A
and
E
E
E
. The circumcircle of
△
B
P
C
\triangle BPC
△
BPC
meets segment
A
B
AB
A
B
at points
B
B
B
and
F
F
F
. Let
I
I
I
and
J
J
J
be the incenters of
△
A
D
E
\triangle ADE
△
A
D
E
and
△
B
C
F
\triangle BCF
△
BCF
, respectively. Segments
I
J
IJ
I
J
and
A
C
AC
A
C
meet at
K
K
K
. Prove that the points
A
,
I
,
K
,
E
A,I,K,E
A
,
I
,
K
,
E
are cyclic.
geometry