Quadrature of Parabola

Theorem
Let $$P$$ be a parabola.

Consider the parabolic segment bounded by an arbitrary chord $$AB$$.

Let $$C$$ be the point on $$P$$ where the tangent to $$P$$ is parallel to $$AB$$.

Let

Then the area $$S$$ of the parabolic segment $$ABC$$ of $$P$$ is given by:
 * $$S = \frac 4 3 \triangle ABC$$

Proof
We consider WLOG the parabola $$y = a x^2$$.

Let $$A, B, C$$ be the points:
 * $$A = \left({x_0, a x_0^2}\right)$$
 * $$B = \left({x_2, a x_2^2}\right)$$
 * $$C = \left({x_1, a x_1^2}\right)$$


 * [[File:ParabolaQuadrature2.png]]

The slope of the tangent at $$C$$ is given by using:
 * $$\frac{\mathrm{d}{y}}{\mathrm{d}{x}} 2 a x_1$$

which is parallel to $$AB$$.

Thus:
 * $$2 a x_1 = \frac {ax_0^2 - a x_2^2} {x_0 - x_2}$$

which leads to
 * $$x_1 = \frac {x_0 + x_2} 2$$


 * [[File:ParabolaQuadrature1.png]]

Historical Note
This proof was given by Archimedes in his book Quadrature of the Parabola, except that he used a different technique to prove that $$\triangle ADC + \triangle CEB = \frac {\triangle ABC} 4$$.