#### English Title

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

#### Enestrom Number

713

#### Fuss Index

322

#### Original Language

Latin

#### Published as

Journal article

#### Published Date

1801

#### Written Date

1778

#### Content 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 2*a*, 2*b*, 2*c*, 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* = (5*qq*–*rr*)/(4*qq*) and *N* = (5*rr*–9*qq*)/(4*rr*). 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 2*a* = 2*qx* + (*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 2*b* = *pr*–*qs* and 2*c* = *pr*+*qs*, and the lengths *g* = (3*pq*+*rs*)/2 and *h* = (3*pq*–*rs*)/2."

#### Original Source Citation

Nova Acta Academiae Scientiarum Imperialis Petropolitanae, Volume 12, pp. 101-113.

#### Opera Omnia Citation

Series 1, Volume 4, pp.290-302.

#### Record Created

2018-09-25