MathDB
Problems
Contests
International Contests
IMO Shortlist
2014 IMO Shortlist
G7
G7
Part of
2014 IMO Shortlist
Problems
(1)
IMO Shortlist 2014 G7
Source:
7/11/2015
Let
A
B
C
ABC
A
BC
be a triangle with circumcircle
Ω
\Omega
Ω
and incentre
I
I
I
. Let the line passing through
I
I
I
and perpendicular to
C
I
CI
C
I
intersect the segment
B
C
BC
BC
and the arc
B
C
BC
BC
(not containing
A
A
A
) of
Ω
\Omega
Ω
at points
U
U
U
and
V
V
V
, respectively. Let the line passing through
U
U
U
and parallel to
A
I
AI
A
I
intersect
A
V
AV
A
V
at
X
X
X
, and let the line passing through
V
V
V
and parallel to
A
I
AI
A
I
intersect
A
B
AB
A
B
at
Y
Y
Y
. Let
W
W
W
and
Z
Z
Z
be the midpoints of
A
X
AX
A
X
and
B
C
BC
BC
, respectively. Prove that if the points
I
,
X
,
I, X,
I
,
X
,
and
Y
Y
Y
are collinear, then the points
I
,
W
,
I, W ,
I
,
W
,
and
Z
Z
Z
are also collinear.Proposed by David B. Rush, USA
IMO Shortlist
geometry