36 KANSAS UNIVERSITY QUARTERLY. 



analytically by means of the process of Geometric Inversion. This 

 theorem which is partially expressed in Wentworth's Geometry, Ex. 

 124, may be stated in full as follows : 



Theorem I. If through a fixed point O on the circumference of a 

 circle whose center is C, a secant GAP ( or AG P) is drawn, meet- 

 ing the diameter BC (produced) in P, so that the exterior segment 

 (or whole secant) AP is equal to the radius, the angle P is ecjual to 

 one-third the angle GCB ; also, if the diameter BC meet the chord 

 AG in P, making AP equal to the radius, the angle APC is etjual to 

 one-third the reflex angle GCB. 



As the proofs of all the constructions explained in this paper are 

 deduced from the principle here establishd, it is desirable to give the 

 following easy demonstration of the three cases of the preceding 

 theorem. 



PROOF OF THEOREM I. 



Case I. When AP is the exterior segment of the secant (See 

 fig. I). 



Let the diameter BCD (produced) meet the secant GAP in P, 

 making AP equal to the radius. 

 To prove <P=^<GCB. 



Draw AC. 

 Then < P = < ACP =Arc AD. 

 Also <P=i Arc GB— f\rc AD=.^<GCB— ^<P. 



.•.<P=.^<GCB. 

 Cor. <AGC=2<P:=|<GCB. 



Case II. When AP is the whole secant (See fig. II). 



