Morley's Theorem states that: If the interior angles of
any triangle are trisected, then the triangle formed
by the intersection
of pairs of adjacent trisectors is always equilateral. The theorem
was not discovered until 1899 by Frank Morley, a professor of
mathematics at Haverford College in Pennsylvania. He discovered
the theorem while investigating algebraic curves tangent to a given number of lines.
Many proofs of Morley's Theorem start with an equilateral triangle
and then proceed to show that given arbitrary positive angles
that sum to 180 degrees, a unique triangle can be constructed such
that the original given triangle is the intersection of the angle
trisectors. However, direct proofs can also be given
It can be shown that using only a straightedge and a compass not every
angle can be trisected. For example a 60 degree triangle cannot
be trisected with only a straightedge and a compass. Thus it is
not possible to construct an arbitrary
Morley equilateral triangle using a straightedge and a compass.
However, if two marks are placed on the straightedge, then
every Morley equilateral triangle can be constructed.
The theorem can be generalized by using angle trisectors
of the interior and exterior angles. This can be done in two ways.
The following way yields three equilateral triangles one lying off of each
To obtain the three new equilateral triangles we trisect two
exterior angles and the other interior angle.
The following way gives one additional central equilateral triangle.
To obtain the additional new central equilateral triangle,
we trisect all three exterior angles.
We now show all five equilateral triangles.
Since Morley's Theorem is true for a triangle, one might wonder
whether it is true for a general quadrilateral. That is perhaps we
always get a rhombus for instance. The answer is no. Even if the
vertices of the quadrilateral lie on a circle, the answer is still