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India IMO Training Camp
2024 India IMOTC
7
7
Part of
2024 India IMOTC
Problems
(1)
Yet another incenter problem
Source: India IMOTC 2024 Day 3 Problem 1
5/31/2024
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
B
<
A
C
AB<AC
A
B
<
A
C
, incentre
I
I
I
, and let
M
M
M
be the midpoint of major arc
B
A
C
BAC
B
A
C
. Suppose the perpendicular line from
A
A
A
to segment
B
C
BC
BC
meets lines
B
I
BI
B
I
,
C
I
CI
C
I
, and
M
I
MI
M
I
at points
P
P
P
,
Q
Q
Q
, and
K
K
K
respectively. Prove that the
A
−
A-
A
−
median line in
△
A
I
K
\triangle AIK
△
A
I
K
passes through the circumcentre of
△
P
I
Q
\triangle PIQ
△
P
I
Q
.Proposed by Pranjal Srivastava and Rohan Goyal
geometry
incenter