MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

54 Mathematical Properties
Figure 2.34: Smallest Inscribed Triangle [57, 62]
This radius is equal to s ·
� 3 √ 3
4π
[161] so that the circular arc has length
.673...s which is much shorter than either the .707...s length of the parallel
bisector or the .866...s length of the altitude.
Property 53 (Smallest Inscribed Triangle). The problem of finding the
triangle of minimum perimeter inscribed in a given acute triangle [62] was
posed by Giulio Fagnano and solved using calculus by his son Giovanni Fagnano
in 1775 [224]. (An inscribed triangle being one with a vertex on each side of
the given triangle.) The solution is given by the orthic/pedal triangle of the
given acute triangle (Figure 2.34 left).
Later, H. A. Schwarz provided a geometric proof using mirror reflections
[57]. Call the process illustrated on the left of Figure 2.34 the pedal mapping.
Then, the unique fixed point of the pedal mapping is the equilateral triangle
[194]. That is, the equilateral triangle is the only triangle that maintains its
form under the pedal mapping. Also, the equilateral triangle is the only triangle
for which successive pedal iterates are all acute [205]. Finally, the maximal
ratio of the perimeter of the pedal triangle to the perimeter of the given acute
triangle is 1/2 and the unique maximizer is given by the equilateral triangle
[158]. For a given equilateral triangle, the orthic/pedal triangle coincides with
the medial triangle which is itself equilateral (Figure 2.34 right).
Property 54 (Closed Light Paths [57]). The walls of an equilateral triangular
room are mirrored. If a light beam emanates from the midpoint of a
wall at an angle of 60 ◦ , it is reflected twice and returns to its point of origin
by following a path along the pedal triangle (see Figure 2.34 right) of the room.
If it originates from any other point along the boundary (exclusive of corners)
at an angle of 60 ◦ , it is reflected five times and returns to its point of origin
by following a path everywhere parallel to a wall (Figure 2.35) [57].