Triangle is inscribed in parabola

Tangents at intersect at forming triangle

Show that the areas of the two triangles are in the ratio

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- Dec 28th 2010, 04:08 AMSorobanA Parabola Property

Triangle is inscribed in parabola

Tangents at intersect at forming triangle

Show that the areas of the two triangles are in the ratio

- Dec 28th 2010, 11:16 AMOpalg
I'll do this for the parabola since I'm more familiar with that. But the proof applies to any parabola.

Take the three points to be The area of the triangle PQR is half the absolute value of Using elementary row operations and expanding down the right-hand column, this comes out to be

The tangent at R is . It meets the tangent at S at the point . Similarly, and The area of the triangle P'Q'R' is half the absolute value of

Thus area(PQR) = 2area(P'Q'R'). - Dec 28th 2010, 11:20 AMred_dog
Let

The slopes of the tangents at are:

The equations of the tangents are:

Solving the systems formed by two of three equations we find the coordinates of the points :

The area of the triangle is

where

Then

But

where

Then

and