Let ABC be a special isosceles triangle with not only AB = AC, but also CB = CD = DA, for a point D on segment AB.
a) Let AB = d and BC = s. Find BD two ways, and from this find an equation relating d and s.
b) Denote by r the ratio d/s. Use your equation from (a) to find r.
c) If angle BAC = x, find (if possible) the measures of all other angles in the figure in terms of x. Then use what you have found to solve for x and thus find the measure of all the angles.