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1969 IMO Shortlist
16
Area of quadrilateral
Area of quadrilateral
Source:
September 29, 2010
geometry
trigonometry
convex quadrilateral
Trigonometric Identities
IMO Shortlist
IMO Longlist
Problem Statement
(
C
Z
S
5
)
(CZS 5)
(
CZS
5
)
A convex quadrilateral
A
B
C
D
ABCD
A
BC
D
with sides
A
B
=
a
,
B
C
=
b
,
C
D
=
c
,
D
A
=
d
AB = a, BC = b, CD = c, DA = d
A
B
=
a
,
BC
=
b
,
C
D
=
c
,
D
A
=
d
and angles
α
=
∠
D
A
B
,
β
=
∠
A
B
C
,
γ
=
∠
B
C
D
,
\alpha = \angle DAB, \beta = \angle ABC, \gamma = \angle BCD,
α
=
∠
D
A
B
,
β
=
∠
A
BC
,
γ
=
∠
BC
D
,
and
δ
=
∠
C
D
A
\delta = \angle CDA
δ
=
∠
C
D
A
is given. Let
s
=
a
+
b
+
c
+
d
2
s = \frac{a + b + c +d}{2}
s
=
2
a
+
b
+
c
+
d
and
P
P
P
be the area of the quadrilateral. Prove that
P
2
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
(
s
−
d
)
−
a
b
c
d
cos
2
α
+
γ
2
P^2 = (s - a)(s - b)(s - c)(s - d) - abcd \cos^2\frac{\alpha +\gamma}{2}
P
2
=
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
(
s
−
d
)
−
ab
c
d
cos
2
2
α
+
γ
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