In Book III, Euclid asks us to believe that "in equal circles angles standing on equal circumferences are equal to one another, whether they stand at the centres or at the circumferences."
The circles shown at the left (code) are "equal," or congruent, and the "equal circumferences" Euclid refers to are represented by the congruent arcs intercepted by the two central angles shown. What this proposition says is that the green central angles are congruent with each other and the red and blue rotating angles ("standing at the circumferences") are congruent with each other, all because they intercept congruent arcs. Cool? Okay, let's take a look at the proof. It's so boring—it's going to knock your socks off.
Another Proof by Contradiction
So, we start by assuming that \(\small\mathtt{\measuredangle{CGB}}\) and \(\small\mathtt{\measuredangle{FHE}}\) are not congruent and that \(\small\mathtt{\measuredangle{CGB}}\) has a greater measure.
By drawing \(\small\mathtt{\overline{GK}}\), let's assume (using I.23, which lets us create a congruent angle) that we have now created a central angle, \(\small\mathtt{\measuredangle{KGB}}\), that is congruent to \(\small\mathtt{\measuredangle{FHE}}\).
Here's the boring part: according to the proposition just before this one (III.26), which I'll get to next time, \(\small\mathtt{\overparen{BK} \cong \overparen{EF}}\) because they are intercepted by congruent central angles. This contradicts our assertion at the beginning that \(\small\mathtt{\overparen{BC}}\) and \(\small\mathtt{\overparen{EF}}\) are equal unless we take arc \(\small\mathtt{BK}\) to be congruent to arc \(\small\mathtt{BC}\), which we'll doodily do.
The rest of the proof uses the Inscribed Angle Theorem, which I've written up here, thankfully, so you have something to read besides this boring proof. Suffice it to say the angles on the circumference are half the measure of the central angle, no matter where they are on the circumference (unless they're inside the central angle, but that's covered at the link as well).
Pareidolic Intellectualism
I'm not sure if you can see this, but if you think of the rotating angles as arms, they look like a golfer swinging her arms, or maybe someone swinging a sword. Or maybe a dance move. I don't know. It seems like these days one can just post a picture of some realia that resembles something vaguely mathematical and pretend they've accomplished something educational. It's the "I F*cking Love Science" syndrome of K-12. So, why not join the fun?
Whew! I managed to snatch something curmudgeonly from the jaws of the purely informative after all!
