Minimum Perimeter of an Inscribed Triangle


For any acute triangle ABC, consruct any inscribed triangle DEF such that D lies on BC, E lies on AC, and F lies on AB.

Click here for 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?


Hint?


Comment:     The triangle DEF with minimum perimeter is the Orthic triangle of 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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