Editing Parabola (section)

== Properties related to Pascal's theorem ==
A parabola can be considered as the affine part of a non-degenerated projective conic with a point <math>Y_\infty</math> on the line of infinity <math>g_\infty</math>, which is the tangent at <math>Y_\infty</math>. The 5-, 4- and 3- point degenerations of [[Pascal's theorem]] are properties of a conic dealing with at least one tangent. If one considers this tangent as the line at infinity and its point of contact as the point at infinity of the ''y'' axis, one obtains three statements for a parabola.

The following properties of a parabola deal only with terms ''connect'', ''intersect'', ''parallel'', which are invariants of [[Similarity (geometry)|similarities]]. So, it is sufficient to prove any property for the ''unit parabola'' with equation <math>y = x^2</math>.

=== 4-points property ===
[[File:Parabel-pk-s.svg|thumb|4-points property of a parabola]]
Any parabola can be described in a suitable coordinate system by an equation <math>y = ax^2</math>.

{{block indent | em = 1.5 | text = Let <math>P_1 = (x_1, y_1),\ P_2 = (x_2, y_2),\ P_3 = (x_3, y_3),\ P_4 = (x_4, y_4)</math> be four points of the parabola <math>y = ax^2</math>, and <math>Q_2</math> the intersection of the secant line <math>P_1 P_4</math> with the line <math>x = x_2,</math> and let <math>Q_1</math> be the intersection of the secant line <math>P_2 P_3</math> with the line <math>x = x_1</math> (see picture). Then the secant line <math>P_3 P_4</math> is parallel to line <math>Q_1 Q_2</math>.

(The lines <math>x = x_1</math> and <math>x = x_2</math> are parallel to the axis of the parabola.)}}

''Proof:'' straightforward calculation for the unit parabola <math>y = x^2</math>.

''Application:'' The 4-points property of a parabola can be used for the construction of point <math>P_4</math>, while <math>P_1, P_2, P_3</math> and <math>Q_2</math> are given.

''Remark:'' the 4-points property of a parabola is an affine version of the 5-point degeneration of Pascal's theorem.

=== 3-points–1-tangent property ===
[[File:Parabel-tk-s.svg|thumb|3-points–1-tangent property]]

Let <math>P_0=(x_0,y_0),P_1=(x_1,y_1),P_2=(x_2,y_2)</math> be three points of the parabola with equation <math>y = ax^2</math> and <math>Q_2</math> the intersection of the secant line <math>P_0P_1</math> with the line <math>x = x_2</math> and <math>Q_1</math> the intersection of the secant line <math>P_0P_2</math> with the line <math>x = x_1</math> (see picture). Then the tangent at point <math>P_0</math> is parallel to the line <math>Q_1 Q_2</math>.
(The lines <math>x=x_1</math> and <math>x = x_2</math> are parallel to the axis of the parabola.)

''Proof:'' can be performed for the unit parabola <math>y=x^2</math>. A short calculation shows: line <math>Q_1Q_2</math> has slope <math>2x_0</math> which is the slope of the tangent at point <math>P_0</math>.

''Application:'' The 3-points-1-tangent-property of a parabola can be used for the construction of the tangent at point <math>P_0</math>, while <math>P_1,P_2,P_0</math> are given.

''Remark:'' The 3-points-1-tangent-property of a parabola is an affine version of the 4-point-degeneration of Pascal's theorem.

=== 2-points–2-tangents property ===
[[File:Parabel-tk-2-s.svg|thumb|2-points–2-tangents property]]

Let <math>P_1 = (x_1, y_1),\ P_2 = (x_2, y_2)</math> be two points of the parabola with equation <math>y = ax^2</math>, and <math>Q_2</math> the intersection of the tangent at point <math>P_1</math> with the line <math>x = x_2</math>, and <math>Q_1</math> the intersection of the tangent at point <math>P_2</math> with the line <math>x = x_1</math> (see picture). Then the secant <math>P_1 P_2</math> is parallel to the line <math>Q_1 Q_2</math>.
(The lines <math>x = x_1</math> and <math>x = x_2</math> are parallel to the axis of the parabola.)

''Proof:'' straight forward calculation for the unit parabola <math>y = x^2</math>.

''Application:'' The 2-points–2-tangents property can be used for the construction of the tangent of a parabola at point <math>P_2</math>, if <math>P_1, P_2</math> and the tangent at <math>P_1</math> are given.

''Remark 1:'' The 2-points–2-tangents property of a parabola is an affine version of the 3-point degeneration of Pascal's theorem.

''Remark 2:'' The 2-points–2-tangents property should not be confused with the following property of a parabola, which also deals with 2 points and 2 tangents, but is ''not'' related to Pascal's theorem.

=== Axis direction ===
[[File:Parabel-ak-s.svg|thumb|Construction of the axis direction]]

The statements above presume the knowledge of the axis direction of the parabola, in order to construct the points <math>Q_1, Q_2</math>. The following property determines the points <math>Q_1, Q_2</math> by two given points and their tangents only, and the result is that the line <math>Q_1 Q_2</math> is parallel to the axis of the parabola.

Let
# <math>P_1 = (x_1, y_1),\ P_2 = (x_2, y_2)</math> be two points of the parabola <math>y = ax^2</math>, and <math>t_1, t_2</math> be their tangents;
# <math>Q_1</math> be the intersection of the tangents <math>t_1, t_2</math>,
# <math>Q_2</math> be the intersection of the parallel line to <math>t_1</math> through <math>P_2</math> with the parallel line to <math>t_2</math> through <math>P_1</math> (see picture).
Then the line <math>Q_1 Q_2</math> is parallel to the axis of the parabola and has the equation <math>x = (x_1 + x_2) / 2.</math>

''Proof:'' can be done (like the properties above) for the unit parabola <math>y = x^2</math>.

''Application:'' This property can be used to determine the direction of the axis of a parabola, if two points and their tangents are given. An alternative way is to determine the midpoints of two parallel chords, see [[Parabola#Corollary concerning midpoints and endpoints of chords|section on parallel chords]].

''Remark:'' This property is an affine version of the theorem of two ''perspective triangles'' of a non-degenerate conic.<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski-planes], p. 36.</ref>

Related: Chord <math>P_1 P_2</math> has two additional properties:
# Its slope is the harmonic average of the slopes of tangents <math>t_1</math> and <math>t_2</math>.
# It is parallel to the tangent at the intersection of <math> Q_1 Q_2 </math> with the parabola.