linguistic:translation, information:classification
Axis direction

The statements above presume the knowledge of the axis direction of the parabola, in order to construct the points Q 1, Q 2 {\displaystyle Q_{1},Q_{2}}. The following property determines the points Q 1, Q 2 {\displaystyle Q_{1},Q_{2}} by two given points and their tangents only, and the result is that the line Q 1 Q 2 {\displaystyle Q_{1}Q_{2}} is parallel to the axis of the parabola.

Let

Then the line Q 1 Q 2 {\displaystyle Q_{1}Q_{2}} is parallel to the axis of the parabola and has the equation x = (x 1 + x 2) / 2. {\displaystyle x=(x_{1}+x_{2})/2.}

Proof: can be done (like the properties above) for the unit parabola y = x 2 {\displaystyle y=x^{2}}.

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 section on parallel chords.

Remark: This property is an affine version of the theorem of two perspective triangles of a non-degenerate conic.