An isosceles triangle is a triangle with two sides equal in length just like the photo below.

Isosceles Triangle

Let the base angle of the isosceles triangle = 2r.

Let (line-AB) = c, (line-BC) = a, (line-AC) = b, Let (line-AD) = d
for isosceles triangle, c = a.

We are asked to solve for b.

S1 = (1/2)cdsin(r)
S2 = (1/2)bdsin(r)

Therefore

(S1/S2) = c/b (1)

or c*c = (b*b)(S1*S1)/(S2*s2) (2)

The perimeter of the triangle

p = 2c + b
The semiperimeter s = (1/2)p = c + (1/2)b

s-a = (1/2)b
s-c = (1/2)b
s-b = c – (1/2)b

the total area of the triangle

S1 + S2 = SQRT[s(s-a)(s-b)(s-c)]
= SQRT{[c + (1/2)b]*(1/2)b*(1/2)b*[c - (1/2)b]}
= (1/2)b*SQRT[c*c - (1/4)b*b] (3)

Square both sides of (3), substitute (2) into (3) and rearrange

The answer:

(line-AC) = b = 2*SQRT[S2*(S1+S2)]/SQRT{SQRT[4*S1*S1 - S2*S2]}

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