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Girls in Math at Yale
2022 Girls in Math at Yale
8
8
Part of
2022 Girls in Math at Yale
Problems
(1)
Girls in Math at Yale 2022 Problem 8: Maximize [DEF]^2
Source:
2/27/2022
Triangle
A
B
C
ABC
A
BC
has sidelengths
A
B
=
1
AB=1
A
B
=
1
,
B
C
=
3
BC=\sqrt{3}
BC
=
3
, and
A
C
=
2
AC=2
A
C
=
2
. Points
D
,
E
D,E
D
,
E
, and
F
F
F
are chosen on
A
B
,
B
C
AB, BC
A
B
,
BC
, and
A
C
AC
A
C
respectively, such that
∠
E
D
F
=
∠
D
F
A
=
9
0
∘
\angle EDF = \angle DFA = 90^{\circ}
∠
E
D
F
=
∠
D
F
A
=
9
0
∘
. Given that the maximum possible value of
[
D
E
F
]
2
[DEF]^2
[
D
EF
]
2
can be expressed as
a
b
\frac{a}{b}
b
a
for positive integers
a
,
b
a, b
a
,
b
with
gcd
(
a
,
b
)
=
1
\gcd (a, b) = 1
g
cd
(
a
,
b
)
=
1
, find
a
+
b
a + b
a
+
b
. (Here
[
D
E
F
]
[DEF]
[
D
EF
]
denotes the area of triangle
D
E
F
DEF
D
EF
.) Proposed by Vismay Sharan
Yale
college