Pappus' trisection using a conchoid
This demonstrates Pappus trisection of an angle, using a neusis construction. The angle to be trisected is the angle between and the positive -axis. Consider the rectangle whose diagonal opposites are and , and extend the line . Draw a line from to cross at and meet this new line at in such a way that the distance is 2. (This is the neusis construction.) This could be implemented with a ruler containing two marks a distance 2 apart, a putting the ruler through so that one mark lies on the line and the other mark lies on the line through . The the angles .
To implement a neusis construction in GeoGebra, we can use a curve known as a conchoid. Given a point , a line , and a distance , a conchoid is the locus of all points for which the distance , where is the point at which the line intersects . In the example here, (the origin), is vertical line through , and . You can check that for any line through the origin, the distance between its intersections with and is 2. If , say, then it is not hard to show that the polar equation of the conchoid is
which can be given parametrically as
So, given a point on the unit circle, we obtain its dstance from the -axis, and then draw the appropriate conchoid. The line which trisects is the line between the origin and the point at which the line through intersects the conchoid.