A definition of a circle, according to The Free Dictionary is [i]a plane curve everywhere equidistant from a given fixed point, the center[/i]. 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 [color=#0000ff]F[/color] and a given line [color=#0000ff]d[/color]. To measure the distance from line [color=#0000ff]d[/color] we consider each point [color=#7d7dff]D[/color] of line [color=#0000ff]d[/color] and take the perpendicular bisector of points [color=#0000ff]F[/color] and [color=#7d7dff]D[/color].

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 [color=#0000ff]F'[/color] and directrix d'. Now let [color=#7d7dff]D' [/color] be an arbitrary point of d', line b the bisector of segment [color=#0000ff]F'[/color][color=#7d7dff]D' [/color] and P' the intersection of the perpendicular to d' in [color=#7d7dff]D' [/color] 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 [color=#7d7dff]D''[/color] (element of d'), for which [color=#7d7dff]D''[/color]P''=P''[color=blue]F'[/color]. Since b is the bisector of [color=#7d7dff]D'[/color][color=#0000ff]F'[/color], also [color=#7d7dff]D'[/color]P''=P''[color=#0000ff]F'[/color] holds. But this means that [color=#7d7dff]D'[/color]P''=[color=#7d7dff]D''[/color]P'', that is in triangle [color=#7d7dff]D'[/color][color=#7d7dff]D''[/color]P'' (which is a right triangle) hypothenuse [color=#7d7dff]D'[/color]P'' and cathetus [color=#7d7dff]D''[/color]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.