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## Question

An ellipse is 1 meter wide at its narrowest point and four times that amount at its widest point.

A triangle is drawn using three points on the ellipse.  How large can it be?

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An equal side triangle in a circle is the largest.
Tilt this inscribed triangle with the circle to grade 1:4 ..
then maybe the projected triangle on vertical plane is
also the largest triangle inscribed in an ellipse.
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If a square of 1 unit area is inclined 1/2 its side backward , it would look half as large in front view..

Polygons, if  are made of thin vertical strips should do too.

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Very close.

What is the area of the ellipse?

Area of projected ellipse(a=.5,b=2): pi(ab)         or pi

is  1/4 Area of tilted circle (r=2): (pi)(2)(2)(1/4) or pi

Projected area of tilted equilateral triangle (h=3) in circle:
is (.25)(3)(3)(tan30) : (.25)(9)(.57735) or 1.29908

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Yah, you got it.
Here's basically just a different wording:

The largest triangle inscribed in a circle is an equilateral triangle.
For a unit circle (area = pi) an inscribed equilateral triangle has area (3/4) sqrt(3) = 1.299038....

If we scale y by f and scale x by 1/f  areas are preserved.
For f=2, a unit circle becomes an ellipse whose axes are 1 and 4 (area = pi).

So the largest triangle inscribed in that ellipse has area (3/4) sqrt(3).

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