112 
Let Simson’s lines of Hy’, He’, and He’, Ha’ and Ha’, Hy’ concur in X, Y, 
and Z respectively. 
The Simson’s lines of A” and H4’, of B’ and Hp)’, and of C’ and H.’ are con- 
jugate, therefore their intersections, u, vy, and w will lie on the nine-point circle 
of rt (, ABC. The Simson’s lines of Ha’, Hp’ and H-’ must pass through Ha, Hn, 
H_- respectively and therefore rt /\ XYZ must have the same nine-point circle as 
/\ ABC. Now since Ha, Hp and He can not be the feet of altitudes they must be 
the midpoints of the sides and therefore u, vy and w must be the feet of altitudes. 
Thus the four points of concurrency are established, namely X, Y, Z and 8S”. 
/\ XYZ, is the ortho- 
8S”, being formed by the intersection of the altitudes of 
center of the same. 
22. If R, Pand Q (Fig. 8) be taken as the midpoints of ares BC, AC and AB, 
respectively, and R’, P’ and Q’ be the points on the circumference opposite R, P, 
and Q, then the Simson’s lines of R, P, and Q will form a (A XYZ, the altitudes 
of which will be the Simson’s lines of R’, Q’ and P’. 
It may be assumed that X YZ is the triangle formed by the intersection of the 
Simson lines of R, P and Q. That the Simson’s line of Q’ is the altitude on side 
XZ may be established thus: 
Z AQv Qe= Z A— Z AQce Qn. 
