Blermions, Cyclic Quadrilaterals, and Crossed Chords

Update: Here is the most recent post about implementing this activity in class. And here is a post about an extension I did on this — involving merblions. I highly recommend reading those! These were some of my favorite days in my geometry class this year!

Last year in Geometry, the other teacher and I weren’t pleased how we introduced the crossed chord theorem. Basically, we ran out of time to come up with a *great* idea and instead had kids measure some things, take the products, and see that the theorem held true. Not our style. We were doing the heavy lifting and kids were making the connection that was served up on a platter.

When we met last week, we decided to rectify this. We brainstormed some general ideas, and I turned those into this activity.

The setup: Kids are so used to looking at “normal” quadrilaterals in geometry. So we thought we’d exploit that. We don’t mention circles. We don’t mention chords.

The TL;DR version: students investigate all quadrilaterals where the diagonals satisfy the property that ac=bd. Students are guided to make a conjecture which we as teachers know will be wrong. Then we show a counter-example to blow their conjecture up. And them bam: they have to try again. Using geogebra and some more encouragement, students discover that all cyclic quadrilaterals satisfy ac=bd. And so the circle emerges out of this investigation of quadrilaterals and diagonals. This is, then, the crossed chord theorem. Which students got at by investigating quadrilaterals. Weird. Now they are in a prime place for wondering why the circle shows up. Proof time!

Here’s the start: we introduce a new type of quadrilateral called a “blermion.” (.docx here)


We had some debate over whether we were giving too much away with this start [1], but we decided we weren’t. (We’re going backwards. The students aren’t deriving the formula. They’re using the formula (which we are calling a “property” of quadrilaterals) to come up with the circle part of the theorem.)

So yeah, we gave kids the ac=bd formula, but in relation to the diagonals of quadrilaterals. And we asked: “which quadrilaterals will this property hold for? We’ll call ’em blermions”

So I ask them to look at the standard quadrilaterals they know — investigating this property using a geogebra sheet — and having them making a conjecture about blermions.


The ggb sheet is here.


So students play on geogebra and come up with some understandings (inductively) about which quadrilaterals are blermions. Then they make a conjecture about all blermions.

This conjecture will fail. Because it is based on students only looking at “nice” quadrilaterals. I want the conjecture to fail. I want to emphasize the point that looking at “nice” examples can often lead to blind spots in your logic.

Students will see it fail when they are asked to drag the four points to specific places (see #5 below). The quadrilateral that results is weird looking. There is nothing that seems special about it. But it does have ac=bd. It is a blermion. Their conjecture about blermions was wrong!

Now students are sent on a chase to find more blermions — and they are encouraged to not just look at “nice” quadrilaterals. They record their results. (If they are stuck, a teacher can have the students fix three points and only drag the fourth point; It turns out you will always be able to drag that point to have ac=bd… and that in fact you can find an infinite number of additional points by doing this dragging of that fourth point.)


At this point, once they have found lots of blermions, students are going to try to make another conjecture about all blermions. I wonder if any student is going to get it. It’s okay if they don’t. At this point, I’m going to have every student plot a different blermion (some “nice” quadrilaterals, but mostly not nice ones). Then I’m going to have them pick any three points and change the color of them. Finally, I’m going to have students go to the “draw a circle with three points” tool, and be surprised by the fact that the circle always goes through that fourth uncolored point.

Why is this good? I hope they don’t get it. Because seeing that every blermion works like this (a circle goes through all four vertices of the blermion) is the key wow factor for kids. It’s strange, because even though I will be giving away this key fact, I think all this play will make this key fact interesting and weird. [2] Once they all see that, they are going to be curious as to how circles even got involved with these quadrilaterals in the first place. And… that is perfect… because then the kids are going to want to know why this happened.

And then we can transition to figuring out how to prove this. Because suddenly the crossed chord theorem is weird and strange and unexpected, and suddenly we kinda want to know why it works!

[1] We had to decide whether students should discover the property ac=bd for crossed chords. Motivating that from a circle and crossed chords was hard. We needed kids to somehow see similar triangles (which felt like we would be giving away too much) or come up with the multiplication idea of the pieces of chords on their own. We had ways to motivate that multiplication, but they weren’t elegant. So we scrapped that.

[2] Here’s the thing. Most things in geometry are presented to students in such a way that their wonderment about the geometric thing is killed. In a proof, the statement to be proved is given up front — and suddenly it isn’t interesting. It might be something really cool, but the exercise around doing the proof doesn’t highlight that. Or — as I’ve blogged before — theorems like the ones involving all the triangle centers… we tell kids to plot the perpendicular bisectors of all three sides of a triangle and they meet at a single point. It isn’t strange and wonderful. They don’t see why that’s weird. They just know we told them to plot the perpendicular bisectors, and they know something will happen because why else would we have them do it? We kill the wonderment of geometry in so many ways.

I want the weirdness and unexpected and unintuitiveness to come back to geometry… that’s where the beauty and curiosity are… and only then have my students work on figuring out why the unexpected happens… and get to the point where the weird and unexpected and unintuitive become obvious and natural. Making the unnatural natural. Yup, that’s the goal. But to do that, you have to first get to the unnatural.



  1. For #5 you could give three points and the location of the intersection, then have them find the fourth point themselves, rather than being told what all four points are.

    I feel like there could be some way here to get kids to think about using the circle themselves. What comes before this? If what comes before this is measuring inscribed angles, you could ask students to measure the angles in blermions. They’ll find the congruent angles, and maybe think this is the same setup as inscribed? At least, they’ll find similar triangles.

    In any case, this is cool and the wonder factor is a big plus, it would be nice to find a way for them to discover the circle instead of being told to try it.

  2. As always Sam you produce gems. An idea for Bowen Kerins question on how to discover the circle, the perpendicular bisectors for each edge of the quadrilateral and the diagonals should all cross at the same point (the center of the circumcircle). I like the approach to using the tool for circle from 3 points. Seems to make the connection quickly without losing in the process.

    I’m a huge fan of this theorem and how it plays out with segments and circles. Not sure if it has any place in your process but here’s a link:

    Thanks for sharing and continuing to challenge us Sam. You’re awesome.

  3. Now that you’ve crossed the million-views mark, I need to get back to stealing good stuff from you (and Brendan) for Geometry. And following you around like a duckling (just you; sorry, Brendan).

    – Elizabeth (@cheesemonkeysf)

  4. I totally agree with your goal to bring back the wonder, weirdness, and curiosity back into geometry. In fact, I’d like to bring those elements back into math in general. I love your lesson activity and how it both incites students’ curiosity and allows the teacher to blow minds! But! I do wonder how such an activity would go over in a classroom where students have been conditioned for so long to being offered math on a platter; they trust that they will be given the right answers and right ideas. If this kind of activity were offered to a classroom of students displaying low levels of self-initiative toward inquiry-based learning, would they feel betrayed that they’re deliberately being led down a path of confusion? Would the weirdness be so unexpected that it’s off-putting? I just completed a practicum (as a preservice teacher) in a classroom where several students easily give up on a task if things don’t go as expected. In this kind of situation, what kind of scaffolding is necessary to build up such students’ mathematical disposition so that a wonderfully thought-out lesson like yours would have positive learning outcomes rather than disappointment/annoyance/demoralization? Maybe this is a naive question coming from someone who doesn’t even have her own classroom yet, but I was just wondering…

    1. Hi Amy… This isn’t a naive question. It’s a great question. First off, this is something I created for the advanced geometry class I teach. So that’s important context. But also: I slowly train my kids from the start of the year to do this sort of work. I wouldn’t just randomly throw this activity in a class where kids aren’t used to it. But they are trained to do this from the very first day.

      To me, it sounds like the question you’re asking is: how do you develop a classroom culture where students are okay with some level of frustration (productive frustration), where they are okay trying things out, where they rely on each other for help, and where they persevere… I don’t think the answer is singular. There are lots of little things you can do — but I think the some important things when doing this are to: (a) celebrate wrong paths/approaches, (b) give a lot of time (you can’t rush this sort of thing), (c) notice when kids are about to turn from productive frustration to unproductive frustration, and have things ready to ask/prompt them to help them get back in the game (or have someone go to other groups to see what other people are doing) and (d) have kids rely on each other and insights that others have — and not to ever be the sole mathematical authority in the room.

      I’m not awesome at this. But I think that one of the best there is is probably Fawn Nguyen… so if you’re trying to develop an inquiry-based classroom culture, her blog might have a wealth of ideas for you!


  5. I really like that you have students lead their own discovery in this lesson. Also I enjoy how you essentially let them come to the wrong conclusion before you show them a counterexample. This allows them to critically think and come to a more specific conclusion. It is a very good way to introduce a proof. How formal would a proof like this be? It has been forever since I have done geometry proofs of this nature.

    1. Right now, this activity is just getting students to make the conjecture that all blermions are cyclic quadrilaterals… The activity doesn’t help them *prove* the crossed-chord theorem in geometry. But later, students do formally prove it using similar triangles. I don’t know how to answer how formal the proof would be. We do make airtight proofs, but we tend to shy away from two column proofs in my class, as I find them to be totally and utterly soul crushing.

  6. I love how interactive and ‘hands-on’ the exploration process is! This is a theorem that’s typically very intimidating and not that interesting for students and you’ve turned it into such an engaging and interesting lesson.

  7. This looks amazing! Can’t wait to implement in my geoemtry class this Spring. Is there an updated link to the ggb activity? Thank you for all you do!

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