Another Catriona Shearer puzzle.
Isosceles I Saw
I have a feeling there is a simpler solution but this is all I’ve got…
The four isosceles triangles are: BCE, CEA, ADC, and BAC.
If the base of triangle BCE is and its two other sides are , the base of triangle BAC is and the two other sides are . They are similar, so , with solution . is the cosine of angle CBE; that angle then is .
As a consequence of the inscribed angle theorem, angles CBE and ADC are supplementary, so the latter is . I didn’t use the fact ADC is isosceles; in fact as long as D is on the circle between A and C, the angle remains the same.