Triangle Inscribing in an Ellipse

$\begin{figure}\begin{center}\BoxedEPSF{EllipseTriangle.epsf scaled 1000}\end{center}\end{figure}$

To inscribe an Equilateral Triangle in an Ellipse, place the top Vertex at , then solve to find the coordinate of the other two Vertices.

$\begin{displaymath} \sqrt{x^2+(b-y)^2}=2x \end{displaymath}$

(1)

$\begin{displaymath} x^2+(b-y)^2=4x^2 \end{displaymath}$

(2)

$\begin{displaymath} 3x^2=(b-y)^2. \end{displaymath}$

(3)

Now plugging in the equation of the Ellipse

$\begin{displaymath} {x^2\over a^2}+{y^2\over b^2}=1, \end{displaymath}$

(4)

gives

$\begin{displaymath} 3a^2\left({1-{y^2\over b^2}}\right)=b^2-2by+y^2 \end{displaymath}$

(5)

$\begin{displaymath} y^2\left({1+3{a^2\over b^2}}\right)-2by +(b^2-3a^2)=0 \end{displaymath}$

(6)

$\displaystyle y$	$\textstyle =$	$\displaystyle {2b-\sqrt{4b^2-4(b^2-3a^2)\left({1+3{a^2\over b^2}}\right)}\over 2\left({1+3{a^2\over b^2}}\right)}$
	$\textstyle =$	$\displaystyle {1-\sqrt{1-\left({1-3 {a^2\over b^2}}\right)\left({1+3{a^2\over b^2}}\right)}\over 1+3 {a^2\over b^2}}\,b,$	(7)

and

$\begin{displaymath} x=\pm a\sqrt{1-{y^2\over b^2}}\,. \end{displaymath}$

(8)