Challenge: Symmetric System of Equations?

Question

Given that a, b, and c are real numbers that satisfy:

a = sqrt(c^2 - (56/5)^2) + sqrt(b^2 - (56/5)^2)
b = sqrt(a^2 - 144) + sqrt(c^2 - 144)
c = sqrt(b^2 - (168/13)^2) + sqrt(a^2 - (168/13)^2).

Find the value of (a+c)/(2b)

Answer 1

This information can be represented in a triangle with sides a, b and c such that the altitudes to the respective sides have lengths 56/5, 12, 168/13.

A = area of triangle:
2A = 56a/5 = 12b = 168c/13
So a = 60b/56, c = 156b/168
a + c = 2b
(a+c) / 2b = 1

Or you could just use software like the first answerer.

Answer 2

a=15
b=14
c=13

(a+c)/(2b)=1