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THEOREM OF THE DAY

Morley's Miracle Let A, B, C be the vertices of a triangle. Let the angle BAC be trisected by lines AB and AC, in that order; similarly let CA and CB trisect angle ACB, and let BC and BA trisect angle CBA. Then the points of intersection, a=AB ∩ BA, b=BC ∩ CB, and c=CA ∩ AC, form the vertices of an equilateral triangle.

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interactive Morley triangles

The applet here, created using David Joyce's Geometry Applet software, provides a test of geometric skill: click on the bottom vertices of the small, red, equilateral triangle to move and resize it. Now, challange a friend to move the vertices of triangle ABC so that the intersection points, a, b and c, exactly fit the equilateral triangle. Space permitting, it is always possible.

Frank Morley (1860-1937) emigrated from England to Pennsylvania in 1887 to teach mathematics at the Quaker college at Haverford and discovered his theorem in 1899, perhaps more miraculous for having never been discovered before than for being surprising or powerful. Morley also excelled at chess, once beating world champion Lasker (also a mathematician). His son, Frank V. Morley, returned to England and became a director, with Geoffrey Faber and T.S. Eliot, of Faber & Gwyer, later Faber & Faber ("Morley, Faber and Eliot would sometimes communicate in exchanges of light verse." John Mullan writing in The Guardian, September 25, 2004)

Web link: www.cut-the-knot.org/ctk/MorleysRedux.shtml

Further reading: Complex Numbers from A to ...Z, by Titu Andreescu and Dorin Andrica, Birkhauser Boston, 2005.

  
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Theorem of the Day is produced and maintained at www.theoremoftheday.org

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