93 
Similarly 
IZ, : 1X, = 8A? : s(sza? — Za?) 
And the same is true of IZ, : 1X. 
The points are therefore collinear. 
Wy SID Be SIGE 
IP, : IM = 8A?: §(SzZa? — Za?) 
IP,:P,M=IP, :IM — IP, = 84? :s(sta? — 2a?) — 8A? = 
=(—a+b+c) (a—b-+-c) (a+ b—c): ska? -- Za? 
—(—a+b+ec)(a—b-+c) (a+b—c). 
The ratio of division is too complex for ordinary use. 
If upon the lines PPa, PP», PPc equal distances from P be taken the triangle 
A,B,C, thus formed is similar to Nagel’s triangle A,B,C,. 
For ZA,PB, =H—C 
And PAB, == 7 PBA, =34(1l — ZA, PB,) #4 C. 
Similarly the 7PB,C, = A. 
And hence 7A,B,C, = 3 A+3C=3(A+ C). 
Likewise 7R,C,A, — 4 (A+ B) 
And £B,A,C,=3(B-+C). 
But these are the angles of Nagel’s triangle and therefore A,B,C, is similar 
to A,B,C. 
P is the symmedian point of the triangle IaIpIc. (Proc. Edinburgh Math. 
Soc., Vol. XI., page 105), 
If through P lines are drawn parallel to the Ialp, IbIc, Icla respectively, the 
six points of intersection with the sides are concyclic. The circle is call Adam’s 
circle. 
