The Renewal Equation

A math odyssey

Groundwork

I spent quite a bit of time today working on the inscribed equilateral triangle problem. I experimented both by hand and using the Desmos geometry tool, which allows you to vary parameters in your drawings. At this point, I’ve come up with a proof that convinces me. There is one step that needs to be formalized and a couple that depend on facts that I know to be true but would like to prove. These include that the sides of a parallelogram are pairwise equal1 and that a triangle is equilateral if and only if all of its angles are equal. The argument also depends heavily on a fact I just learned today, conjecturing it based on experimentation and then confirming it with a Google search: inscribed angles subtended by the same arc of a circle are equal (as shown below). I’m not sure whether I will be able to prove that or not. My education in geometry focused almost exclusively on lines and triangles, and I know little of circle geometry.

Two Subtended Angles

(As readers may have guessed, I’m feeling much better today.)

  1. To do. ↩︎
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Responses

  1. Tim McL

    Quit reading now if a hint about circles is not welcome.

    Let arc $A$ be the arc subtended by your angles $x$. Let $\alpha$ be the measure of the angle at the center of the circle subtending arc $A$. Then $x=\alpha/2$. Want to prove this? Then since you know about triangles…

    Reply
    1. Olly

      Thank you for the hint (and the warning). Those are actually somewhat stylized arcs with hashmarks, not exes. Nevertheless, the first time I looked back at the diagram after a while away from it, I thought, “x? what’s x?” :-/

      Reply
      1. Tim McL

        Ah, right. I see that now. How perfectly clear, once you see that it’s clear. If you want to see how Euclid proved that angles on the circumference are half the size of angles from the center to the same arc, it’s Prop III.20 ff in the Elements.

        If you want a bit more Timmish confusion, I had forgotten what Ptolemy’s Theorem was, and when you had a post with the title, “Triangles, Triangles,” I thought, “Are triangles the right tool? Isn’t there something about cyclic quadrilaterals? I wonder if it would be helpful?” (For those who, like me, don’t know what Ptolemy’s Theorem is, yes, there is a theorem about cyclic quadrilaterals. It’s not all that useful in proving Ptolemy’s Theorem, though, because it IS Ptolemy’s Theorem.)

        I’m impressed that as I’m writing this, you have multiple solutions to the equilateral triangle problem. You’re making more progress than I think I would have.

        Remember the Geometry Problem of Doom, which we both messed with?

        Reply
        1. Olly

          I do remember that problem. I never did solve it. I think I know what notebook I have it written in, though. Perhaps I should pick it up again.

          Reply
  2. Kim J

    Better is better.

    Reply
    1. Olly

      That’s what we in the math business call a tautology.

      Reply
  3. yerpa

    as in “A taught Olly, gee!” which you are thanks to such as Tim…

    Reply
    1. Olly

      Groan.

      Maybe if I keep going to the gym, I will also become a taut Olly.

      Reply
  4. Angles and Arcs – The Renewal Equation

    […] also ready to prove the proposition about inscribed angles from my post Groundwork. It turns out that proposition falls right out of another that Tim suggested starting with when I […]

    Reply

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