MathDB
Problems
Contests
National and Regional Contests
Malaysia Contests
Malaysian IMO Training Camp
2024 Malaysian IMO Training Camp
8
8
Part of
2024 Malaysian IMO Training Camp
Problems
(1)
Proving three lines concurrent
Source: Own. Malaysian SST 2024 P8
9/5/2024
Given a triangle
A
B
C
ABC
A
BC
, let
I
I
I
be the incenter, and
J
J
J
be the
A
A
A
-excenter. A line
ℓ
\ell
ℓ
through
A
A
A
perpendicular to
B
C
BC
BC
intersect the lines
B
I
BI
B
I
,
C
I
CI
C
I
,
B
J
BJ
B
J
,
C
J
CJ
C
J
at
P
P
P
,
Q
Q
Q
,
R
R
R
,
S
S
S
respectively. Suppose the angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
meet
B
C
BC
BC
at
K
K
K
, and
L
L
L
is a point such that
A
L
AL
A
L
is a diameter in
(
A
B
C
)
(ABC)
(
A
BC
)
.Prove that the line
K
L
KL
K
L
,
ℓ
\ell
ℓ
, and the line through the centers of circles
(
I
P
Q
)
(IPQ)
(
I
PQ
)
and
(
J
R
S
)
(JRS)
(
J
RS
)
, are concurrent.Proposed by Chuah Jia Herng & Ivan Chan Kai Chin
geometry