This demonstrates Pappus trisection of an angle, using a [i]neusis[/i] construction. The angle to be trisected is the angle [math]\alpha[/math] between [math]OA[/math] and the positive [math]x[/math]-axis. Consider the rectangle [math]OBAD[/math] whose diagonal opposites are [math]O[/math] and [math]A[/math], and extend the line [math]BA[/math]. Draw a line from [math]O[/math] to cross [math]AD[/math] at [math]C[/math] and meet this new line at [math]F[/math] in such a way that the distance [math]CF[/math] 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 [math]O[/math] so that one mark lies on the line [math]AD[/math] and the other mark lies on the line through [math]BA[/math]. The the angles [math]AFC = COD = \alpha/3[/math]. To implement a neusis construction in GeoGebra, we can use a curve known as a [b]conchoid[/b]. Given a point [math]P[/math], a line [math]L[/math], and a distance [math]d[/math], a conchoid is the locus of all points [math]Q[/math] for which the distance [math]RQ = d[/math], where [math]R[/math] is the point at which the line [math]PQ[/math] intersects [math]L[/math]. In the example here, [math]P=O[/math] (the origin), [math]L[/math] is vertical line through [math]A[/math], and [math]d=2[/math]. You can check that for [i]any[/i] line through the origin, the distance between its intersections with [math]L[/math] and [math]J[/math] is 2. If [math]AB = OD = d[/math], say, then it is not hard to show that the polar equation of the conchoid is [math]r = 2 + d \sec(\theta)[/math] which can be given parametrically as [math]x = 2 \cos(\theta)+d, y = 2 \sin(\theta) + d \tan(\theta)[/math] So, given a point [math]A[/math] on the unit circle, we obtain its dstance [math]d[/math] from the [math]y[/math] -axis, and then draw the appropriate conchoid. The line which trisects [math]\alpha[/math] is the line between the origin and the point at which the line through [math]BA[/math] intersects the conchoid.