Is there a simpler way to show this?
Is this proof original?
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Although this requires extending a line segment, I think it's easier to see that a $180^\circ$ line is divided in two (without algebraic rearrangment) - and the inscribed $90^\circ$ angle is right there, half of the line.
Also, the corresponding angle used is arguably simpler than the alternate interior angle in the question.
Regarding presentation, the lighter coloured angles are more legible. I think moving the inscribing angle nearer to the center would be better, to make the red and blue angles closer in measure, and therefore more equally visible. But I'd have to redo most of it, and it's a time-consuming nightmare making angles colours go between the segments you want in geogebra/classic (it sometimes prefers the other end of the segment, as if extended).