Circle theorem - "same segment" query

Question

In the diagram, we see an example of one circle theorem which in many textbooks is stated as "angles in the same segment are equal".

Why do we refer to $CBE$ and $CDE$ as angles in the same segment? For example, $BCD$ and $BED$ are also 'in' the pink segment, but they are not equal to $57.47$.

Answer 1

A name only has to remind us of what it means; it doesn't have to be a complete theorem statement

Within the pink segment, there are lots and lots of angles. As well as $BCD$ and $CBE$, there are $COB$, $BCE$, and we can construct even more points, lines and angles at will. They can't all be equal to each other. So the phrase "angles in the same segment" could not be a complete description of what the result is about. We have to understand from context that when we say "the angles-in-the-same-segment theorem" we mean "you know, that one theorem we learned about, where you draw a line across the circle and then a bunch of triangles all on the same side with that as the base, and their third points on the circumference, and all those top angles turn out to be equal." It's similar to saying "Pythagoras" as a shorthand for the well-known theorem, often attributed to Pythagoras, concerning the side lengths of right-angled triangles.

But there is more to say about this name!

The phrase "In a circle, angles in the same segment are equal to one another" comes from Euclid's Elements (Book 3, Proposition 21) so it has quite the pedigree in geometry -

᾿Εν κύκλῳ αἱ ἐν τῷ αὐτῷ τμήματι γωνίαι ἴσαι ἀλλήλαις εἰσίν.

(Greek text from Johan Ludvig Heiberg, Euclidis Elementa; Leipzig: B. G. Teubner, 1883-1888)

The English phrase appears in this exact form in several translated editions over the past centuries, as well as in textbooks and related material. Euclid was held in great esteem historically and it's only comparatively recently that we've stopped teaching from his works directly. It is not too surprising to find this wording preserved in the educational context.

Euclid has also been critiqued, on the grounds of both paedogogy and rigour, and perhaps this Proposition is an example. He often states a result that we might want to be a bit more precise, because his text assumes that you will get the right idea from reading his proof and the surrounding material. In this case, his proof of 3.21 follows straight from 3.20, which showed that (my emphasis!)

In a circle, the angle at the center is double that at the circumference, when the angles have the same circumference base.

(translation is Richard Fitzpatrick, 2007)

The bolded phrase is omitted in 3.21 but "ought" to be there from context, given that Euclid's brief proof is - using your notation - that $CBE$ and $CDE$ are both equal to half of $COE$, and hence are equal to each other. The proof does not work unless we can apply 3.20, which we only proved with the bolded caveat.