Apollonius’ theorem
Apollonius’ theorem on the parabola
Apollonius of Perga stated around 200 BC:
Any three tangents of a parabola are divided by their points of intersection and points of tangency into segments of like proportion.
(H. Dörrie, 100 great problems of elementary mathematics: their history and solution, New York: Dover Publications, 1965.)
Thus, according to figure below it is as follows |AP|/|PS|=|PO|/|OQ|=|SQ|/|QB|.
Geometric proof
To prove this Apollonius’ theorem on the parabola we take advantage of the dynamic geometry features of GeoGebra and modify the geometric proof provided by Dörrie in his aforesaid book. The resulting ’visual and dynamic’ proof is shown in figure below.
The main method of this proof is the parallel projection of segments of tangents on lines parallel to the directrix (see lines numbered 1, 2 and 3). Due to the validity of another property of parabola that says that "the point of intersection of two parabola tangents lies on a parallel to the parabola axis, passing through the midpoint of the chord connecting two points of tangency" (see the Dörrie's book), six segments of different lengths are projected on segments of only three distinct lengths, k (red), l (blue) and k+l (see the line No. 2). Then the proof of the theorem, namely in its form |AP|/|PS|=|PO|/|OQ|=|SQ|/|QB|=k/l, arises from a suitable rearrangement of these projections. Moving with the three lines parallel to the directrix by dragging points 1, 2 and 3 try to develop arguments for this geometric proof.