566 
PROBLEMS AND SOLUTIONS. 
[383 
It is now clear that the right-hand side will be independent of if only 
sin (0' — 0") (a cos 0 +b sin 0) + sin (0" — 0) (a' cos 6' + b' sin 0') 
+ sin (0 - 0') {a" cos 0" + b" sin 0") = 0, 
sin (0' — 0") (— a sin 0 + b cos 0) + sin (0" — 0) (— a' sin 0' + b' cos 0') 
+ sin (0 — 0') (— a' sin 0" + b" cos 0") = 0 ; 
equations which show that, given the form of the triangle and the centres of two of 
the circles, the centre of the third circle (in the porismatic case) is a determinate 
unique point : and the theorem is thus proved. 
[Vol. i. pp. 137—141.] 
1273. (By the Editor [W. J. Miller, B.A.].)—In a given triangle let three 
triangles be inscribed, by joining the points of contact of the inscribed circle, the 
points where the bisectors of the angles meet the sides, and the points where the per 
pendiculars meet the sides; then will the corresponding sides of these three triangles 
pass through the same point; also the triangle formed by the three points of inter 
section will be a circumscribed co-polar to the original triangle, and the pole will be 
on the straight line in which the sides of the given triangle meet the bisectors of its 
exterior angles. 
1. Solution by Professor Cayley. 
The theorem is, in fact, included in the following more general 
Theorem. Let the points 0, O', 0", ... lie on a conic circumscribed about a 
triangle ABC; then first the polars of the points 0, O', 0", ... in regard to the 
triangle (see Note at the end of the Solution) pass through a fixed point Î2. And 
secondly, if by means of the point 0, joining it with the vertices A, B, C, and taking 
the intersections of these lines with the sides BG, G A, AB, respectively, we form a 
triangle inscribed in the triangle ABC; and the like for the points O', 0",...; the 
corresponding sides of the inscribed triangles meet in three points forming a triangle 
circumscribed about the original triangle ABC, and such that the lines joining the 
corresponding vertices of the last-mentioned two triangles meet in the point fl. 
But, in order to see that the proposed theorem 1273 is in fact included under 
the foregoing more general one, it is necessary to state the following 
Subsidiary Theorem. Consider a conic inscribed in the triangle ABC, and 
passing through the points I, J. 
Take 0 the pole of the line IJ in regard to the conic ; 0' the point of inter 
section of the lines joining the vertices of the triangle with the points of contact 
on the opposite sides respectively ; 0" the point of intersection of the lines Al, Bm, Cn, 
where i is a point on BG such that the lines lA, IBG, II, IJ form a harmonic pencil, 
and the like for the points to and n respectively.