The orthic triangle DEF is the one of least perimeter inscribed in an accute angled triangle ABC.

Reflect triangle ABC successively 5(6) times along its sides, each time on a different side. Take a side, BC say, and watch its position after each reflexion. After the sixth reflexion it returns to its initial (parallel to it) position. This because the angles by which it revolves sum up to 2pi. The same happens with a side, IK say, of an inscribed triangle IJK. Thus following consecutive sides on the reflected IJK's we obtain a broken line (aquamarine) II''' of length equal to twice the perimeter of IJK. In particular, because of the characteristic property of the (yellow)

Besides there is no other triangle with perimeter equal to that of the orthic. This follows from the fact that the middle of II''' is on A'B' only for the case of F' corresponding to the orthic triangle. We have though infinite many

The arguments used above prove analogously the following property for polygons inscribed in triangles:

Given an integer multiple of 3, n=3*k > 0, and a polygon p = P

The solution given here, due to H.A.Schwarz, has some implications for the billiard ball problem, see the references below. Another solution to Fagnano's problem is given in Fagnano2.html .

Orthic.html

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