Last time (here), we talked about III.27 from The Elements, which told us that "in equal circles angles standing on equal circumferences are equal to one another, whether they stand at the centres or at the circumferences." Here we'll go over the proof of III.26, which is different from III.27 in ways not easy to access based on just the description: "in equal circles equal angles stand on equal circumferences, whether they stand at the centres or at the circumferences." So . . . let's take a look.
We start by drawing some things and making some assumptions. First, Circles \(\small\mathtt{ABC}\) and \(\small\mathtt{DEF}\) are congruent, the central angles (\(\small \mathtt{\color{green}{\measuredangle{BGC}}}\) and \(\small\mathtt{\color{green}{\measuredangle{EHF}}}\)) are congruent, and the inscribed angles (\(\small\mathtt{\color{blue} {\measuredangle{BAC}}}\) and \(\small\mathtt{\color{blue}{\measuredangle{EDF}}}\)) are also congruent. What we want to show is that the minor arc \(\small\mathtt{BC}\) is congruent to minor arc \(\small\mathtt{EF}\).
Chain of Begats
Euclid proceeds by drawing \(\overline{\small\mathtt{BC}}\) and \(\overline{\small\mathtt{EF}}\), which I've already done above. He then concludes that \(\overline{\small \mathtt{BC}}\) and \(\overline{\small\mathtt{EF}}\) are congruent by reasoning as follows (using SAS, or I.4, which we covered here):
- Circles \(\small\mathtt{ABC}\) and \(\small\mathtt{DEF}\) are congruent, so their radii are congruent.
- This means that \(\overline{\small\mathtt{BG}}\) is congruent to \(\overline{\small\mathtt{FH}}\) and \(\overline{\small\mathtt{CG}}\) is congruent to \(\overline{\small\mathtt{EH}}\).
- The central angles (\(\small\mathtt{\color{green}{\measuredangle{BGC}}}\) and \(\small\mathtt{\color{green}{\measuredangle{EHF}}}\)) are congruent by assumption.
- Thus, \(\overline{\small\mathtt{BC}} \cong \overline{\small\mathtt{EF}}\) via SAS.
The Peril of Working Backwards
The sequence of the propositions in the Elements is important, which we'll feel right now when I tell you that the next step in this proof relies on a previous theorem, III.24. That proof, which we'll have to get to later, shows that when two circles have arcs ("segments," actually) that lie on congruent line segments, those two arcs are also congruent. (There's also some weirdness in the definition of "similar segments" used in that proof [and this one] which we may want to think about later too.)
For our purposes, this means that major arc \(\small\mathtt{BC}\) is congruent to major arc \(\small\mathtt{EF}\). And since the two entire circles are congruent by assumption, this means that the minor arcs must be congruent too.
So, in III.27, we showed that the angles are equal when we are provided a given that the intercepted arcs are equal. In this proof, we show that the arcs are equal when we are provided a given that the angles are also.
Sequence Is Important to Think About, Not Necessarily to Follow
Order is something I definitely think a lot about when it comes to designing curricula. And our public conversation about sequencing in mathematics education strikes me as incredibly naive. What I have to listen to more often than not is that students "need X to do Y." I mentioned above that a proof of III.24 was used to construct a proof of III.26, but it does not necessarily follow that III.24 is required to create this proof (it might be required, though). Needing X to do Y is often defended on similarly shallow evidence. Usually, the foundation for this argument is simply the sequence that the arguer is used to. Or a shallow understanding of a topic.
While it is certainly true that some topics must appear before others, we should also think about what benefits to learning playing around with order can bring. At the moment, what seems to drive almost all of our sequencing decision-making is tradition, ignorance, and fear.
Image credit: pratanti