42 



KANSAS UNIVERSITY QUARTERLY 



The straight line PB inverts into a circle through the left hand 

 vertex, and therefore with its center on the directrix. The line PO 

 is not changed by inversion and therefore becomes a focal radius of 

 the hyperbola. 



Since the magnitude of an angle is not changed by the process of 

 inversion, the following Inverse Theorem is now completely estab- 

 lished. • 



Theorem II. If a circle is described, having for its center any point 

 on the directrix of a hyperbola whose eccentricity is 2, passing 

 through the nearer focus and farther vertex, and intersecting the 

 nearer branch of the curve in P, the angle which it makes with the 

 shorter focal radius drawn to P is equal to one-third the angle it 

 makes with the axis at the farther vertex. 



THIRD MElHOn. 



Let BCD (fig.VIin 

 be the given angle. 

 Take CF=^CA, any 

 convenient length. 

 With F as focus and .A 

 the more remote ver- 

 tex, construct the hy- 

 perbola whose eccen- 

 tricity is 2. 



With C as a center, 

 describe a circle pass- 

 ing through A and F, 

 and intersecting the 

 hyperbola in P, P'. and 

 P". 



Then PC trisects the 

 angle ACF. 



Proof. The vertex C lies on the directrix CN, which is the per- 

 pendicular bisector of AF. 



Draw the tangents PG and AK. 

 Then < FPG=J <FAK. (Theorem II.) 



. •. <FCP=^ <ACF. 

 Likewise <P'CF^^ the reflex angle ACF. 

 The direct proof* of this method is also very simple. 

 Draw PI:h perpendicular to CN. 



*This solution has recently been given bv Mr. L. C. Duncan of Holtmi. Kan . who 

 worked n out directly and Independently. It Is included in this paper because it naturally 

 belongs to the series of solutions here presented. "■ . • 



