An illustration of Euclid III.21


Rotating lines

The diagram can be viewed in two ways.

First way: The open dot moves on a circle. The solid dots are two fixed points on that circle. The lines connecting the moving point to the fixed points always form the same angle — if you follow the open dot around, you will see that the shape of the X does not change. (This fact is a variation on Euclid III.21: "In a circle the angles in the same segment equal one another.")

Second way: The solid dots are two fixed points. Through each of them passes a line. Each line rotates about its fixed point at the same rate. The open dot is the intersection of the lines. As the lines rotate, their point of intersection traces out a circle. (This fact is a converse to III.21, not stated by Euclid, but well-known.)

2006 July 21


Mail Steven: steven@amotlpaa.org