For any

acutetriangle ABC, consruct anyinscribed triangleDEF such that D lies on BC, E lies on AC, and F lies on AB.

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herefor a GSP sketch of an inscribed triangle as picture above.

Is there a unique inscribed triangle with the minimum perimeter?

What if the triangle ABC was a right triangle? an obtuse triangle?

Comment: The triangle DEF with minimum perimeter is the

Orthic triangleof triangle ABC. That is, D, E, and F are at the follow of the perpendiculars from A, B, and C, respectively. When triangle ABC is a right triangle, the Orthic triangle degenerates to an altitude segment. When triangle ABC is obtuse, the Orthic triangle is not inscribed because parts of it are outside of the triangle.

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