114 
Now, Z PRR’ — 90° — PR’R. 
0° = 1A 2B) 
= -4'C. 
*. arc BQ=arc PR’, and so 2 ADP= / CAR’. Hence AT is || PQ and 
.*. Simson’s line of R is | to PQ. In like manner we may prove Simson’s line of P 
| to RQ, and Simson’s line of Q | to RP. 
25. The Simson’s lines of Ha”, Hv”, He” (Fig. 9) with respect to -/\ 
HaHpHe, inscribed in the nine-point circle, form a rt. /\ XYZ, the altitudes of 
which are the Simson’s lines of Ma, Mp and M.- with respect to the same rt. /\. 
H, the orthocenter of /, ABC, is the incenter of (| HaHpHe, for 7s AH-H 
and AH»pH being right 7s, A, He, H and Hp» are concyclic with AH as diameter. 
