The Renewal Equation

A math odyssey

Return to the Triangle of Doom

I felt better today. I worked for about an hour on exercises in Analysis with an Introduction to Proof and for another half hour on a question about logarithms that occurred to me while doing one of the exercises. I also spent quite a bit of time on a puzzle I call the Triangle of Doom. (It is more widely known as the Hardest Easy Geometry Problem.) I first encountered and worked on it in 2006, but before today I hadn’t looked at it in a long time.

The puzzle is to find the the value of $x$ in the figure below without using trigonometry.1 (Image source)

Doom Plain

I didn’t solve the Triangle of Doom today, but I had some ideas.

The measures of many of the angles are easy to find using familiar geometry: vertical angles, supplementary angles, and the fact that the angles of a triangle sum to 180 degrees.

Doom With Angles Measures

Adding the fact that if two angles of a triangle are equal, then the sides that subtend them are also equal, reveals that the whole triangle is isosceles and that it also contains two smaller isosceles triangles, as highlighted below in blue and green. I suspect these isosceles triangles will have some role in the solution.

Doom Isosceles Triangles 1

Something I’m not sure the significance of is that there also appear to be two similar triangles in the figure, as highlighted below in yellow. I cannot yet prove that these are similar—that would be to solve the puzzle—but tests using the Desmos geometry tool left me nearly sure. (You can check out my interactive drawing here for however long the link lasts.) The relationship might be a coincidence, though, since it is not preserved when three isosceles triangles overlap in the same way but with different angle measures.

Doom Similar Triangles
  1. To do. ↩︎
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Responses

  1. Yerpa

    I was confused looking at yhe yellow triangle figure, to which I was referred by a statement that the yellow triangle in those positions were similar in the first example. The angle labels on said figure contradicts that similarity. I see that that contradiction was about your later questioning of the generality of the first statement, so I am no longer confused. But when you publish your book, you might want to polish this.

    Reply
    1. Olly

      They are similar with reflection.

      Reply
  2. Tim McL

    I spent an unbelievable amount of time on this problem after you introduced me to it. I don’t remember whether I finally solved it or whether I finally gave up. (Maybe the latter?) I remember working hard to place the $20^\circ$ angles on the circumference of a circle to make 9-gon, which I thought was very beautiful but which turned out as far as I could tell to be completely unhelpful.

    I don’t think it’s a spoiler (especially since I no longer remember how to do the problem) to say that the solution requires drawing one or two additional (unexpected to me) line(s). That’s the reason I thought of the problem a few days ago. Your solution to the problem in “The Proof At Last” also involved a crafty extra line. I looked at that solution and was reminded of the new line(s) in the Geometry Problem of Doom and thought, “Wow. If Olly can do that, I wonder if she could even do the Geometry Problem of Doom?”

    I think the GPoD is a lot trickier. Don’t get hung up and spend the next year on it. But it’s fun to see it again, and it might be good practice to mess about with it while also moving forward in some of the other threads you have going.

    So much fun.

    Reply
    1. Olly

      Yes, the caution not to let the Triangle absorb too much of my time is a good one. I only worked on it a little today.

      Reply

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