Parabolas Inscribed in Shared Tangent Lines

Theorem
Let the function $\map f x = A x^2 + B x + C_1$ be a curve embedded in the Euclidean Plane.

Let $\map {y_1} x$ be the equation of the tangent line at $\tuple {Q, \map f Q}$ on $f$.

Let $\map {y_2} x$ be the equation of the tangent line at $\tuple {-Q, \map f {-Q} }$ on $f$.

Then there exists another function $\map g x$ also embedded in the Euclidean Plane defined as:
 * $\map g x = -A x^2 + B x + C_2$.

with:
 * tangent lines $\map {y_3} x$ being the equation of the tangent line at $\tuple {Q, \map g Q}$ on $g$

and:
 * $\map {y_4} x$ being the equation of the tangent line at $\tuple {-Q, \map g {-Q} }$ on $g$.

so that the tangent lines $y_3$ and $y_4$ inscribe $\map f x$ and the tangent lines $y_1$ and $y_2$ inscribe $\map g x$.

Proof
The tangent line at $\tuple {Q, \map f \Q}$ on $f$ is defined as:


 * $\map {y_1} x = \paren {2 A Q + B} x + b_1$

where $2 A Q + B$ is the slope of the tangent line on the point $\tuple {Q, \map g Q}$ on $f$. Substitute in the coordinates of the point $\tuple {Q, \map g Q}$ to $y_1$ and solve for $b_1$.

This will reveal the $y$-intercept of $y_1$:

Continue by following the same steps for $y_2$ which is defined:


 * $\map {y_2} x = \paren {-2 A Q + B} x + b_2$

where $-2 A Q + B$ is the slope of the Tangent line at the point $\tuple {-Q, \map f {-Q} }$ on $f$.

Substitute in the coordinates of the point $\paren {-Q, \map f {-Q} }$ to $y_2$.

Use these values to solve for $b_2$, and this will reveal the $y$-intercept of $y_2$:

The $y$-intercepts of both $y_1$ and $y_2$ have been shown to be equivalent.

Since $b_1 = b_2$ redefine this value as $b$.

The distance between $b$ and $C_1$ is $\size {C_1 - b}$.

Let $\map g x = -A x^2 + B x + C_2$.

Then the Tangent line at the point $\tuple {Q, \map g Q}$ on $g$ is defined as:


 * $\map {y_3} x = \paren {-2 A Q + B} x + b_3$

where $-2 A Q + B$ is the slope of the tangent line at $\tuple {Q, \map g Q}$ on $g$.

Solve for $b_3$ using the same methods used for $y_1$ and $y_2$.

This will reveal the $y$-intercept of $y_3$:


 * $b_3 = A Q^2 + C_2$

The result also follows for the Tangent line $\tuple {-Q, \map g {-Q} }$ on $g$ which is defined:


 * $y_4 = \paren {-2 A Q + B} x + b_4$

Solving for $b_4$ yields the result:


 * $b_4 = A Q^2 + C_2$

The $y$-intercepts of both $y_3$ and $y_4$ have been shown to be equivalent.

Notice that the derivatives of $f$ and $g$ satisfy:

Then it must be true that:
 * $y_1 = y_4$ and $y_2 = y_3$

and the functions $y_1$, $y_2$, $y_3$, and $y_4$ share the same $y$-intercept.

Redefine this the $y$-intercepts of the tangent lines as $b$.

Solve for $C_2$ to determine the vertical translation of $\map g x$:

Therefore the function:


 * $\map g x = -A x^2 + B x - \paren {2 A Q^2 + C_1}$

will have tangent lines equivalent to the tangent lines on $\map f x$ at the points $\tuple {Q, \map f Q}$, and $\tuple {-Q, \map f {-Q} }$.