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But, since Q, Qc, A, and Q» are concyclic, Z AQnQe— Z AQQe, which is 
measured by 4 are AQ’. : 
Also, since Q, B; Qa, and Qe are concyclic, Z BQQc—=180°— 7 BQaQe. 
-. Z QeQaC= Z BQQc, which is measured by 4 arc BQ’. But arc BQ’ = 
arc AQ’. | 
(A.) .*. Z AQnQec= Z CQaQe. 
But Z AQpQc= Z A— Z AQ-Qpd, and 
Z CQaQe= Z B+ Z BQcQa. 
= Z B+ Z AQcQp. 
Now the Simson’s line of Q’ is | to QnQa the Simson’s line of Q, at Qc, and 
Z AQcPp = Z Band Z BQ-Ra= Z A. And from (A.) we know that Z 
QrQcPr = Z QaQcRa, .*. Z ProQeH’, = Z RaQ-H’ and thus Simson’s line of Q’ 
is proven to be the internal bisector of Z Qe in rt. A, RaPpQe. In like manner 
the Simson’s line of R’ may be proven to be the internal bisector of 7 Ra, and Sim- 
son’s line of P’, the internal bisector of 2 Pa of same triangle. Since the Sim- 
son’s lines of Q, R, and P are | to these internal bisectors at the vertices of the 
Z\, they must be the external bisectors of the same angles. But we know that the 
internal and external bisectors of the /s of a triangle concur in four points. 
Thus X, Y, Z and H’ are each points of concurrency of three Simson’s lines. 
H’, of course, is the ortho-center of (A XYZ. 
23. /\ XYZ has the same nine-point circle as /\ ABC and is therefore 
inscribable in the same sized circle as rt. /\ ABC. 
That it has the same nine-point circle follows easily from the fact that three 
points that must lie on its nine-point circle, the feet of its altitudes, are coincident 
with three points on the ABC nine-point circle, the midpoint of its sides. Since 
three points determine a circle, their nine-point circles must be identical. 
24, The Simson’s line of P is || to RQ, Simson’s line of R is || to PQ, and 
Simson’s line of Q is || to RP. 
The Simson’s line of R will be | to Simson’s line of R’, so let us prove Sim- 
son’s line of R’ | to PQ’. 
Now, the Simson’s line of R/’ will be | to the line isogonal conjugate to AR’ 
which we call AT. Then let us prove AT || to PQ. 
To be || Z ADP must = Z CAR’. 
Z ADP is measured by 3 arc (AP-+ BQ). 
Z CAR’ is measured by 3 are CR’. 
Arc AP=arc CP, and 3 are BQ measures 3 / C. 
8—ScIENCE. 
