Editing Parabola (section)

=== 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.