Geometric mean in geometry
Large version
of the image
For
△
A
B
C
,
I
is the midpoint of
B
C
, point
D
is in line segment
A
C
such that
C
D
=
3
A
D
and point
E
is in line segment
A
B
such that
[
B
I
E
]
=
[
C
I
D
]
×
[
A
D
E
]
.
If the ratio E B A E can be expressed as c a − b , then find a + b + c .
(The figure is not drawn to scale)
Note: [ X Y Z ] denotes the area of triangle X Y Z .
The answer is 114.
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1 solution
L
e
t
A
E
=
X
,
a
n
d
E
B
=
1
L
e
t
a
r
e
a
Δ
A
B
C
=
S
A
B
C
S
B
I
E
=
2
1
∗
1
+
X
1
∗
S
A
B
C
.
S
C
I
D
=
2
1
∗
4
3
∗
S
A
B
C
.
S
A
D
E
=
4
1
∗
1
+
X
X
∗
S
A
B
C
.
∴
(
2
1
∗
1
+
X
1
∗
S
A
B
C
.
)
2
=
(
2
1
∗
4
3
∗
S
A
B
C
.
)
∗
(
4
1
∗
1
+
X
X
∗
S
A
B
C
)
.
⟹
1
+
X
1
=
8
3
∗
X
∴
3
X
2
+
3
X
−
8
=
0
.
S
o
l
v
i
n
g
t
h
e
q
u
a
d
r
a
t
i
c
X
=
6
−
3
+
9
+
9
6
=
6
−
3
+
1
0
5
.
B
u
t
E
B
A
E
=
X
.
∴
a
+
b
+
c
=
1
1
4
.
Same solution . Nice a good presentation .
Same solution.Hats off to your great presentation.