# Isocseles I Saw

Another Catriona Shearer puzzle.

13.
Isosceles I Saw

All 4 triangles are isosceles. What’s the angle?

Spoiler!

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 $b$ and its two other sides are $s = 1$, the base of triangle BAC is $1$ and the two other sides are $1+b$. They are similar, so $b/1 = 1/(1+b)$, with solution $b = (\sqrt{5}-1)/2$. $b/2$ is the cosine of angle CBE; that angle then is $72^\circ$.

As a consequence of the inscribed angle theorem, angles CBE and ADC are supplementary, so the latter is $108^\circ$. 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.