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 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.

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*Related*

I just realized the whole thing could be part of a regular pentagram, so if there is an answer, it’d have to be that.