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?
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.