109 
Draw P’H’ ( Fig. 7), letting it cut BC in U’. Then, from above proposition, 
HU’ | to Simson’s line of P’.. Also HU || to Simson’s line of P. 
.. Z (S, 8S’)= Z (HU, HU’). Now, H is the center of similitude of the 
circumcircle and the nine-point circle. Draw P’H, letting it cut the Simson line 
of P’ at D’. 
Then P and D, P’ and D’ and H’ and Ha are corresponding points. 
.°. HaD and HaD’ and H’P and H’P’ are || lines, whence 7 DHaD’ = 
mene — / UU. 
15. If P and Q (Fig. 1) be the extremities of a diameter and R and KR’ two 
other opposite points such that PR and QR’ are | to Simson’s line of P and PR’ 
and QR are | to Simson’s line of R, then the Simson’s line of R is parallel to PQ 
and Simson’s line of R/ is | to PQ. 
Since the angle between the Simson’s lines of two points is equal to an angle 
inscribed -in the arc between them, we know that 2 ZXY = Z YQZ. Also 
ZX || QY. 
.. QZXY is a parallelogram, and XY, the Simson’s line of R, is || to PQ. 
Then it follows that Simson’s line of R is | to PQ, since it is conjugate to Simson’s 
line of R. 
16. If ES and FS ( Fig. 3) are the Simson’s lines of opposite points on the 
circumcircle, and EF be any other Simson’s line, then T’/E—=T’F = T’S, where 
T’ is the center of the last named Simson’s line. 
Z T’ED= Z T’SD (a previous proposition ). 
eek hos, In like manner we can show 1’F TS. .:. ’E=T’F: 
The Simson’s lines of opposite points on the circumcircle are said to be 
conjugate. 
17. The are between the vertices of two Simson’s lines (not conjugate) is 
twice as large as the arc between their centers. For ET’S is an isosceles / 
endef S == 2 750. =) Bat bl’ Ss 871"8. .°. are SS’=2 are TT, Now 
suppose T’S is less than R (it never can be greater), then S” could be 
another point on the nine-point circle such that T’S’’—T’S and 4 ES’ F would 
also be aright 7. It is thus evident that there are always two pairs of conjugate 
Simson’s lines passing through E and F. 
The limit of EF is 2R. 
For when S and 8” coincide at'8’”, T’S’’”—T’E=T’F=r. In this case 
we have but one pair of conjugate Simson’s lines. 
18. If two Simson’s lines, SD and S’D’, which are not conjugate, cut a 
third Simson’s line, T’S’, at equal distances, E and F, from its center, T’, then 
