At the end
of the 19th century, mathematician Frank Morley discovered that the
trisectors of each angle of any triangle meet in three new points that form a new
triangle that is always an equilateral triangle. The fact that this property holds for all
triangles, including scalene triangles, was so unexpected that it became to be
known as Morley’s Miracle. (An interactive
applet for Morley’s Miracle can be found at here.)
The proof is
rather lengthy, but only requires a knowledge of high school trigonometry (including
the law of sines, the law of cosines, Pythagorean identity, and sine and cosine
addition and subtraction formulas). The
basic outline of the proof is to first use the law of sines on ∆ACE to express
c1 as a function of ∠A, ∠C, and b; then use the law of sines on ∆ABF
to express b3 as a function of ∠A, ∠B, and c; and finally use the law of
cosines on ∆AEF to show that the value of a2 is symmetrical when
expressed in terms of the sides, angles, and area of ∆ABC. The same calculations used to find a2
can then be used to find b2 and c2, which by their
symmetrical equations make them all equal to each other, showing that ∆DEF is an
equilateral triangle. The full proof can
be found here.
Another surprising
aspect of Morley’s Miracle is that the theorem cannot be extended to polygons
with more sides than triangles. Trisecting
the four angles of a quadrilateral does not always produce a rhombus, and
neither does quadrisecting the four angles.
It also does not appear that Morley’s Miracle can be extended to higher
dimensions. Trisecting the four angles
of a tetrahedron does not always produce a regular tetrahedron.
Quadrilateral with Trisected Angles
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