90 
Tue Pornt P anp Some oF Its PROPERTIES. 
By Rogsert J. ALEY. 
P is the point of concurrence of the lines drawn from the vertices of a triangle 
to the points of contact of the inscribed circle with the sides. It has been called 
the Gergonne Point. The ratios of the distances of the point P from the sides are 
ws : me y is (Aley, Contributions to Geom. of the Triangle, p. 
10 (10) ). From these ratios the actual distances of the point from the sides is 
easily found to be 
2A (s—b)(s—c) 
rts ~ ad (s—a)(s—b) 
__-2A(s —c) (s—a) 
hdcae Sate Te =H} 
PP. __ 2A4(s — a) (s — b) 
c= (s —a)(s— hb) 
P and Q (Nagel’s Point) are isotomic conjugates and they are collinear with 
Z, the isotomic conjugate of I (incentre) (Ibid., page 8, III). 
P, (the isogonal conjugate of P), Z, the isogonal conjugate of Z and K are 
collinear (Ibid., page 13, IV). 
P,, I and M are collinear. 
The ratios of P, are a(s— a) : b(s —b) : e(s —c) (Ibid., p. 3, 81). 
From these the actual distances of P, from the sides is readily found to be 
> cao oar a) 
Pair la — S=a == Sas 
2Ab(s — b) 
PiPi» = Sya? Sai 
2Ac(s — 
Pi Pic = ee | Ibid. » page 14, (7) | . 
It is well known that 
fa 
s 
1 ee 
8 
eee 
8 
The ratios of M are a(— a?-+- b? + c?) : b(a®? — b? +c?) : c(a® + b? — c?). 
