Geometry Musings #6 Trisecting an Angle


The trisection of an angle using only a compass and straight edge was a famous problem in Ancient Greece. It was proven to be impossible by Pierre Wantzel in 1837, and proven possible using origami later (anyone know by whom?). Origami isn’t allowed though for Euclidean Geometry.

However there are some angles you can trisect using Euclidean Geometry. I demonstrated how to construct an angle of 30 degrees a few weeks ago, and perpendicular lines before that, so trisecting 90 degrees is clearly possible.

Here’s another fun way, which uses the construction of a pentagon. If we join the vertices of a pentagon, we trisect the internal angles.

So here we go:

Step 1: Arbitrary Circle


Step 2: Perpendicular to the diameter (use two identically sized largeish circles, or arcs if you want to be neat and tidy):



Step 3: Draw in a perpendicular bisector of the radius (CO in this pic)


There’s a lot of lines now, here’s a crude “what I did in the pic above”:


Step 4: Draw a circle with diameter CO:



Step 5: Draw a straight line from B going through D, and thus creating point E intersecting the circle, and a second new point at the further edge of the new circle, point F. Then draw in two circles, one with radius DE and one with radius DF



Now we can just join up the points to make up a pentagon as shown (click for enlargement):


Now draw lines to join the vertices, and you’ve just trisected a 108 degree angle:


It’s a big mess, so I did a neater version without most of the construction lines:



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