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Triangle Ellipse Inscribing

EllipseTriangle

To inscribe an equilateral triangle in an ellipse, place the top polygon vertex at (0,b), then solve to find the (x,y) coordinate of the other two vertices.

sqrt(x^2+(b-y)^2)=2x
(1)
x^2+(b-y)^2=4x^2
(2)
3x^2=(b-y)^2.
(3)

Now plugging in the equation of the ellipse

(x^2)/(a^2)+(y^2)/(b^2)=1,
(4)

gives

3a^2(1-(y^2)/(b^2))=b^2-2by+y^2
(5)
y^2(1+3(a^2)/(b^2))-2by+(b^2-3a^2)=0
(6)
y=(2b-sqrt(4b^2-4(b^2-3a^2)(1+3(a^2)/(b^2))))/(2(1+3(a^2)/(b^2)))
(7)
=(1-sqrt(1-(1-3(a^2)/(b^2))(1+3(a^2)/(b^2))))/(1+3(a^2)/(b^2))b,
(8)

and

x=+/-asqrt(1-(y^2)/(b^2)).
(9)

See also

Castillon's Problem, Ellipse, Equilateral Triangle

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Cite this as:

Weisstein, Eric W. "Triangle Ellipse Inscribing." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TriangleEllipseInscribing.html

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