Source: TW
This is a fantastic story of an Iyengar from Bangalore who, in 1909, found one of the earliest known solutions to one of the most beautiful problems in geometry.
His name was Mandyam Tondanur Naraniengar, and he taught mathematics at Central College, Bangalore.
The problem was Morley’s theorem…..
Take any triangle and cut each of its angles into 3 equal parts. Each side of the triangle now has 2 of these cutting lines running nearest to it, 1 from each end.
Where the 2 lines nearest a side meet, mark a point. Do this for all 3 sides, and the 3 points always form a perfect equilateral triangle inside. Every triangle, every time.
Frank Morley had found this around 1899, and in 1908 it appeared as an open problem in the Educational Times, a London journal.
Naraniengar’s solution appeared in the 1909 collected volume…
His idea went backwards. Instead of starting with the given triangle and forcing an equilateral triangle out of it, he built the equilateral triangle 1st, from the trisectors of 2 of the angles.
Then he proved that the lines joining it to the 3rd vertex must be the trisectors of the 3rd angle. So the triangle he built was exactly Morley’s triangle. A small lemma about equal chords in a circle does the heavy work.
A colleague at Central College, V. Ramaswami Aiyar, remembered that in those days “we discussed nothing but mathematics.” Naraniengar went on to become one of the founding joint secretaries of what grew into the Indian Mathematical Society, and later its president.
In 1967, Coxeter and Greitzer included Naraniengar’s proof in Geometry Revisited, more than 50 years after it appeared in the Educational Times.
Sources: M. T. Naraniengar’s solution, Mathematical Questions and Their Solutions from the Educational Times, New Series, Vol. 15 (1909), p. 47; H. S. M. Coxeter and S. L. Greitzer, Geometry Revisited (1967), Section 2.9; V. G. Tikekar, “Morley’s Miracle - Part I”, At Right Angles, Vol. 3, No. 2 (2014); V. Ramaswami Aiyar’s presidential address to the Indian Mathematical Society (1926).
Photo 1: M. T. Naraniengar (1871-1940)
Photo 2: The original printed problem, highlighted in yellow.
Photo 3: Naraniengar’s actual published solution. His name, construction and conclusion are highlighted in yellow, with the original diagram beside them.