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Topic: Inscribed ellipse (Read 612 times) |
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Obob
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Inscribed ellipse
« on: Sep 28th, 2009, 3:55pm » |
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Given any triangle, show that there is an inscribed ellipse meeting the three midpoints of the triangle.
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Benny
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Re: Inscribed ellipse
« Reply #1 on: Sep 28th, 2009, 4:10pm » |
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Isn't it the Marden's theorem ?
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If we want to understand our world — or how to change it — we must first understand the rational choices that shape it.
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Obob
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Re: Inscribed ellipse
« Reply #2 on: Sep 28th, 2009, 4:12pm » |
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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.
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| « Last Edit: Sep 28th, 2009, 4:21pm by Obob » |
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Grimbal
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Re: Inscribed ellipse
« Reply #3 on: Sep 29th, 2009, 12:26am » |
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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.
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