I was reading an article this evening and saw a footnote to a book by Arthur Baragar [1]. This caught my eye because he was my officemate at UT for a year.

I found his book on Archive.org and was surprised to see “The Star Trek Lemma” in the table of contents. What could this be?

It’s a theorem that goes back to Euclid that applies to an angle formed by connecting a point to two other points on a circle. The theorem says “The measure of an inscribed angle is half the measure of the arc it subtends.” But why call it the Star Trek lemma? Quoting Arthur:

In the spirit of Euclid, we will refer to this theorem as the Star Trek lemma because of the figure associated with the statement of the theorem. … Before Star Trek, as far as I know, this theorem had no name, though some might call it Euclid III.20, which is its proposition number in Euclid’s Elements (Book III, Proposition 20).

Here is my reconstruction of the figure given in the book.

The lemma says that ∠BAC is half of ∠BOC.

[1] Baragar, Arthur (2001), A Survey of Classical and Modern Geometries: With Computer Activities, Prentice Hall