One of the angles of an isosceles triangle is 108 degrees, find the ratio of the lengths of two bisectors of unequal angles.

Since the triangle is isosceles, the bisector is the height, and the median, and divides the base in half, and the angles at the base are equal to each other: (180 -108) / 2 = 72/2 = 36 degrees each. Let the base be equal to c, and the lateral the sides are a, the bisector to the base is la, and to the side is lb.
Consider a triangle composed of the bisector, la and part of the side b. According to the sine theorem: la / (a ​​/ 2) = (sin 36) / sin (180-36-90) = sin 36 / sin 54 = sin 36 / sin ( 90 – 36) = tg 36. (1)
For a triangle from the bisector lb, side a, and part b, the theorem of sines:
lb / a = sin 36 / sin (180-36-18) = sin 36 / sin 126 = sin 36 / sin (90 + 36) = -tg 36 (2).
Divide (1) by (2), we get:
la / lb: (2) = 2 * (la / lb) = tg36 / (- tg36) = – 1.
And the ratio of the bisectors in modulus is la / lb = 1/2.
That is, the bisector of an obtuse angle is 2 times shorter than other bisectors.