Motivation
- Author:
- Zoltán Kovács
A definition of a circle, according to The Free Dictionary is a plane curve everywhere equidistant from a given fixed point, the center. Now by constructing the tangent line in a point of the circle we find that the tangent is perpendicular to the radius. When considering the trace of the tangent lines, we find that the union U of the tangents of a circle is the whole plane except the disc inside the circle.
Now let us consider the same figure from another point of view. Having the set U we may be interested of a curve such that its tangents are the lines of U. Such a curve can be the given circle, but that it is the only possible curve (that is, the question has a unique answer) is not straightforward.
On the other hand, each element of U is equidistant from the center of the circle because distance is defined by measuring orthogonal projection. This idea leads us to define a parabola by considering the set U' of lines being equidistant from a given point F and a given line d. To measure the distance from line d we consider each point D of line d and take the perpendicular bisector of points F and D.
We have already mentioned in the previous section that usual definition of a parabola p' is that it is the locus of points P' which are equidistant from focus F' and directrix d'. Now let D' be an arbitrary point of d', line b the bisector of segment F'D' and P' the intersection of the perpendicular to d' in D' and b. A well-known property of b that it is the tangent of parabola p' in point P'. To prove that, consider the following figure.
Let us assume that b is not a tangent of parabola p' in point P'. Then there is another point P'' on b, also element of the parabola, that is b is a secant line of p'. Assumably this point P'' was created by foot point D'' (element of d'), for which D''P''=P''F'. Since b is the bisector of D'F', also D'P''=P''F' holds. But this means that D'P''=D''P'', that is in triangle D'D''P'' (which is a right triangle) hypothenuse D'P'' and cathetus D''P'' have the same length, which is impossible. This contradiction ensures that b is a tangent, not a secant.
Thus we proved that these two different definitions of a parabola (i.e. the classical by using locus, and this second one which uses the concept of trace of the bisector) is equivalent, that is they define the same parabola.