Area of an equilateral triangle.?

Question

There is a point inside of an equilateral triangle that is 4 inches away from the bottom left vertex, 5 inches away from the bottom right vertex and 3 inches away from the top vertex.

What is the area of the triangle?

Answer 1

There must be a nicer way...
Nevertheless, let's wade in:

Consider points of the equilateral triangle A=(0,0) = bottom left, B=(s,0) = bottom right and C=(s/2,s√3/2) = top. The area of this triangle is (s²√3)/4.

The set of points 4 units away from A are given by: x² + y² = 4²
The set of points 5 units away from B are given by: (x-s)² + y² = 5²
Subtract the first equation from the second to get (Eq3):
s² - 2sx = 9
or x = (s²-9)/(2s)

Now the set of points 3 units away from C are given by:
(x-s/2)² + (y - (s√3)/2)² = 3²
This expands to:
x² - xs + y² - ys√3 + s² = 9
Subtract this from the second equation to get:
ys√3 - xs = 16
or
xs = ys√3 - 16

Now let's plug this into Eq3 (s² - 2sx = 9):
s² - 2(ys√3 - 16) = 9
or
2ys√3 = s² + 23
simplifying to
y = (s² + 23)/(2s√3)

Now that we have both x and y in terms of s, we plug them both into the first equation (x² + y² = 16):
((s² - 9)/(2s))² + ((s² + 23)²/(2s√3))² = 16
Multiply through by 4*3s² to clear denominators:
3(s² - 9)² + (s² + 23)² = (3s²)4³
Now let u = s² to simplify notation just a bit:
3(u²-18u+81) + (u² + 46u + 529) = (3u)4³
Combine a few terms:
4u² - 8u + 772 = (3u)4³
u² - 2u + 193 = 48u
u² - 50u + 193 = 0

Use the quadratic formula, (-b+/-√(b²-4ac))/2a, to find that
u = (50+/-√(50²-4*193))/2
= 25 +/- √(625-193)
= 25 +/- √432
= 25 +/- 12√3

Recall that the mystery inner point is 5 units away from a vertex. The altitude must be at least that great so that
(s√3)/2>5
Thus s >10/√3 so that s² > 100/3
Thus, we take the positive sign from the above equation for u.

That is
Area = (s²√3)/4 = (u√3)/4 = (25√3 + 12*3)/4 = 9 + (25√3)/4

Answer 2

root3/4*side