pure geometry

The following geometry problem is simple to state but challenging to solve!

Pure geometry or Euclidean geometry is a body of theorems and corollaries logically derived from certain axioms and postulates as presented in Euclid’s Elements. Later geometers, both Greek and others, have added to this. Occasionally some algebra is brought in but not trigonometry. Abraham Lincoln is said to have read the Elements just for the reasoning.

In the March 2013 issue of AtRiA, the following result had been stated in the article on ‘Harmonic Triples’: Let ΔPQR have ∡P = 120˚. Let PS be the bisector of ∡QPR, and let PQ = a, PR = b, PS = c; then 1 / a + 1 / b = 1 / c. It had been proved using trigonometry, and the question was asked: Is there a proof using ‘pure geometry’? We give just such a proof here...
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