Triangle is inscribed in parabola
Tangents at intersect at forming triangle
Show that the areas of the two triangles are in the ratio
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').
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