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        riddles >> medium >> Inscribed ellipse
        
(Message started by: Obob on Sep 28^th, 2009, 3:55pm)

Title: Inscribed ellipse
Post by Obob on Sep 28^th, 2009, 3:55pm

Given any triangle, show that there is an inscribed ellipse meeting the three midpoints of the triangle.

Title: Re: Inscribed ellipse
Post by BenVitale on Sep 28^th, 2009, 4:10pm

Isn't it the [hide]Marden's theorem[/hide] ?

Title: Re: Inscribed ellipse
Post by Obob on Sep 28^th, 2009, 4:12pm

I didn't know it was called that, but yes, it appears to be a part of that theorem. However, there is a very nice proof of just this part of the theorem, that is interesting to look for.

Title: Re: Inscribed ellipse
Post by Grimbal on Sep 29^th, 2009, 12:26am

[hide]
With an equilateral triangle, it is obvious. You can inscribe a circle that will touch all midpoints.

For any other triangle, make an affine transform that map the equilateral triangle to the other triangle. The same transformation applied to the circle gives an ellipse with the properties you want.
[/hide]