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Math Forum » Discussions » Math Topics » geometry.pre-college.independent

Topic: construction of triangle of given perimeter, given point and angle
Replies: 16   Last Post: Jun 10, 2011 12:58 PM

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John Conway

Posts: 2,238
Registered: 12/3/04
Re: minimum perimeter
Posted: Nov 7, 1997 6:01 PM
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On Fri, 7 Nov 1997, Joshua Zucker wrote:

> The question:
> Given an acute triangle, find the inscribed triangle (with one point
> on each side of the original triangle) of minimum perimeter.
>
> The answer:
> The triangle formed by the feet of the altitudes of the original
> triangle has the smallest perimeter.
>
> The proof:
> I don't know yet.
>
> --Joshua Zucker

But I do! This is a well-known theorem, and there's an
easy proof by reflecting the triangle in its sides so as to
convert the problem into a shortest path one (think of light
bouncing around inside a triangle of mirrors).

John Conway

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