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3-points–1-tangent property

Let P 0 = (x 0, y 0), P 1 = (x 1, y 1), P 2 = (x 2, y 2) {\displaystyle P_{0}=(x_{0},y_{0}),P_{1}=(x_{1},y_{1}),P_{2}=(x_{2},y_{2})} be three points of the parabola with equation y = a x 2 {\displaystyle y=ax^{2}} and Q 2 {\displaystyle Q_{2}} the intersection of the secant line P 0 P 1 {\displaystyle P_{0}P_{1}} with the line x = x 2 {\displaystyle x=x_{2}} and Q 1 {\displaystyle Q_{1}} the intersection of the secant line P 0 P 2 {\displaystyle P_{0}P_{2}} with the line x = x 1 {\displaystyle x=x_{1}} (see picture). Then the tangent at point P 0 {\displaystyle P_{0}} is parallel to the line Q 1 Q 2 {\displaystyle Q_{1}Q_{2}}.(The lines x = x 1 {\displaystyle x=x_{1}} and x = x 2 {\displaystyle x=x_{2}} are parallel to the axis of the parabola.)

Proof: can be performed for the unit parabola y = x 2 {\displaystyle y=x^{2}}. A short calculation shows: line Q 1 Q 2 {\displaystyle Q_{1}Q_{2}} has slope 2 x 0 {\displaystyle 2x_{0}} which is the slope of the tangent at point P 0 {\displaystyle P_{0}}.

Application: The 3-points-1-tangent-property of a parabola can be used for the construction of the tangent at point P 0 {\displaystyle P_{0}}, while P 1, P 2, P 0 {\displaystyle P_{1},P_{2},P_{0}} 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.