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?


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.

Return to the EMAT 4600/6600 Page