proof of Morley’s theorem
To simplify some formulas, let us denote the angle , or 60 degrees, by . Denote the angles at , , and by , , and respectively, and let be the circumradius of . We have . Applying the sines law to the triangle ,
Combining that with the identity
Using the cosines law now,
But we have
whence the cosines law can be applied to those three angles, getting
Remarks: It is not hard to show that the triangles , , and are isoscoles.
By the sines law we have
This implies that if we identify the various vertices with complex numbers, then
provided that the triangle has positive orientation, i.e.
I found Letac’s proof at http://www.cut-the-knot.org/triangle/Morley/index.shtmlcut-the-knot.org, with the reference Sphinx, 9 (1939) 46. Several shorter and prettier proofs of Morley’s theorem can also be seen at cut-the-knot.
|Title||proof of Morley’s theorem|
|Date of creation||2013-03-22 13:45:44|
|Last modified on||2013-03-22 13:45:44|
|Last modified by||mathcam (2727)|