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."
Publication:

Originally published in Nova Acta Academiae Scientarum Imperialis Petropolitinae 12, 1801, pp. 101113

Opera Omnia: Series 1, Volume 4, pp. 290  302
 Reprinted in Commentat. arithm. 2, 1849, pp. 294301 [E713a]
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