E713 -- Investigatio trianguli in quo distantiae angulorum ab eius centro gravitatis rationaliter exprimantur

(An investigation of a triangle in which the distances of the angles from the center of gravity of it may be expressed rationally)


Summary:

"This memoir was made to please a small number of Amateurs of Indeterminate Analysis, and contains a very beautiful solution to the problem announced in the title. Here it is in a few words: Let the sides of the desired triangle be 2a, 2b, 2c, and the lengths drawn from their midpoints to the opposite angles be f, g and h, respectively. Take two numbers q and r, and find M = (5qq - rr)/(4qq) and N = (5rr - 9qq)/(4rr). Reduce the fraction ((M - N)2 - 4)/(4(M + N)) to its lowest terms and name the numerator x and the denominator y, and you will have the side 2a = 2qx + (M - N)qy, and the line f = rx - (1/2)(M - N)ry. Make p = x + y and x = x - y, and you will have the sides 2b = pr - qs and 2c = pr + qs, and the lengths g = (3pq + rs)/2 and h = (3pq - rs)/2."

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