Geometry Musings #2 Trisection of a line

I find it a little theraputic to recreate Euclidean Geometry with a ‘compass and straight rule’ rather than with Geogebra (which is no slight on Geogebra, it’s amazing!).

Anyway, I thought I’d share today’s effort – the trisection of a line.

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Here’s my line. Very interesting so far no?

OK, so now I need an arbitrary point above the line, which is going to be the centre of a circle, whose radius is from that arbitrary point, to the end of the line:

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(this is not the only way to do this by the way, there are LOADS of ways).

Now I’m going to keep my compass fixed, and draw a second circle, with centre at the point of intersection between my new line and the top of the circle:

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By joining the top point of intersection of my second circle to the end point of my original line (the one we’re trisecting), I get an important line:

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To trisect the original line, I now need a second and third line parallel to this new one, coming from the centre point of both circles.

Now, I could just drag the ruler, but this is EUCLIDEAN GEOMETRY, so I can’t do that. I need to construct parallel lines appropriately. (my diagrams are kind of rubbish by the way, I’m sure Euclid was rather more careful and didn’t put giant dots everywhere).

So, to construct parallel lines going through a point, I used this method (it’s complicated and needs diagrams!), and repeated it twice. Once for a parallel line going through the centre of one circle, and again for the other circle. The points of intersection with my original line trisect it. Hurrah!

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