Calculate the area of an triangle that is inscribed into an ellipse such that the elliptical sectors are of equal size.

geometry triangles conic-sections

Solution 1:

Thanks to the hint of markvs, we can simplify the problem as follows:

  • We set $a=b$ and have a circle (with radius $b=a$) instead of the ellipse.
  • The elliptical sectors remain equally sized.
  • This leads to the (new/projected) triangle being equilateral.
  • The area of an equilateral triangle inscribed in a circle with radius $b$ is $\frac{3}{4}\sqrt{3}b^2$.

Stretching the area back to the ellipse (along the x-axis) leads to the area:

$$T_{\triangle ABC}=\frac{a}{b}\cdot\frac{3}{4}\sqrt{3}b^2$$

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