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2007 Harvard-MIT Mathematics Tournament
18
18
Part of
2007 Harvard-MIT Mathematics Tournament
Problems
(1)
2007 Guts #18: Diagonals of Convex Quadrilateral
Source:
6/22/2012
Convex quadrilateral
A
B
C
D
ABCD
A
BC
D
has right angles
∠
A
\angle A
∠
A
and
∠
C
\angle C
∠
C
and is such that
A
B
=
B
C
AB=BC
A
B
=
BC
and
A
D
=
C
D
AD=CD
A
D
=
C
D
. The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at point
M
M
M
. Points
P
P
P
and
Q
Q
Q
lie on the circumcircle of triangle
A
M
B
AMB
A
MB
and segment
C
D
CD
C
D
, respectively, such that points
P
P
P
,
M
M
M
, and
Q
Q
Q
are collinear. Suppose that
m
∠
A
B
C
=
16
0
∘
m\angle ABC=160^\circ
m
∠
A
BC
=
16
0
∘
and
m
∠
Q
M
C
=
4
0
∘
m\angle QMC=40^\circ
m
∠
QMC
=
4
0
∘
. Find
M
P
⋅
M
Q
MP\cdot MQ
MP
⋅
MQ
, given that
M
C
=
6
MC=6
MC
=
6
.
geometry
circumcircle