In Which I Am Disturbed By A Triangle Theorem That Isn’t Morley’s Somehow


I’ve been reading Alfred S Posamentier and Ingmar Lehmann’s The Secrets of Triangles: A Mathematical Journey. It is exactly what you’d think: 365 pages, plus endnotes and an index, describing what we as a people have learned about triangles. It’s almost enough to make one wonder if we maybe know too many things about triangles. I admit letting myself skim over the demonstration of how, using straightedge and compass, to construct a triangle when you’re given one interior angle, the distance from another vertex to its corresponding median point, and the radius of the triangle’s circumscribed circle.

But there are a bunch of interesting theorems to find. I wanted to share one. When I saw it I felt creeped out. The process seemed like a bit of dark magic, a result starting enough that it seemed to come from nowhere. Here it is.

Start with any old triangle ABC. Without loss of generality, select a point along the leg AB (other than the vertices). Call that point P. (This same technique would work if you put your point on another leg, but I would have to change the names of the vertices and line segments from here on. But it doesn’t matter what the names of the vertices are. So I can suppose that I was lucky enough that whatever leg you put your point P on I happened to name AB.)

Now. Pick the midpoint of the leg AB. This median is a point we’ll label S.

Draw the line PC.

Draw the line parallel to the line PC and which passes through S. This will intersect either the line segment BC or the line segment AC. Whichever it is, label this point of intersection R.

Draw the line from R to P.

The line RP divides the triangle ABC into two shapes, a triangle and (unless your P was the median point S) a quadrilateral.

The punch line: both shapes have half the area of the original triangle.

I usually read while eating. This was one of those lines that made me put the fork down and stare, irrationally angry, until I could work through the proof. It didn’t help that you can use a technique like this to cut the triangle into any whole number you like of equal-area wedges.

I’m sure this is old news to a fair number of readers. I don’t care. I haven’t noticed this before. And yes, it’s not as scary weird magic as Morley’s Theorem. But I’ve seen that one before, long enough ago I kind of accept it.

Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

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