Editing Parabola (section)

=== 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.