The usual proof of inscribed circle theorem uses the fact that the sum of interior angles is equal to exterior angle. I've formulated a little different approach, although the underlying concept is still to use isosceles triangle properties. I want to know whether my proof has anything original or not. The proof is,
A,B and C are thrre vertices of the triagnle and O is the circumcenter. By the property of isosceles triangle we have $\delta = \delta _1, \alpha = \alpha _1$ and $\beta = \beta _2$
Now, $$(\angle A - \alpha)+(\angle C - \beta) = 180 - \angle AOC$$ $$\implies \angle AOC = \left( 180-(\angle A + \angle C) \right) + (\beta + \alpha)$$ $$\implies \angle AOC = \angle B + \angle B = 2 \angle B$$
- So the question is, have I done something original?