91 



The Cayleyan Cubic. 

 By C, A. Waldo and John A. Newlin. 



The U.se of the Bicycle Wheel in Illustrating the Principles 



OF THE GrYROSCOPE. 



By Chas. T. Knipp. 



(Published in the Physical Review, Vol. XII, No. 1, January, 1901. 



The Cyclic Quadrilateral. 



By J. a Gregg. 



PROBLEM. 



The opposite sides of a quadrilateral FGHI inscribed in a circle, when 

 produced, meet in P and Q; prove that the square of PQ is equal to 

 the- sum of the squares of the tangents from P and Q to the circle.— 

 No. 80, page 470, Phillips and Fisher's Geometry. 



SOLUTION. 

 (See Fig. I.) 



On PO and QO as diameters draw circles (centers S and T) and cutting circle 

 O in C, D, E and K. QK and PD are tangent to O. Through the points Q, F 

 and G draw a circle cutting PQ in A. Then Z PHG = Z GFI = Z QAG 

 .•. ZPAG 18 the supplement of Z PHG and PAGH is cyclic, and 



PQ. PA = PF. PG = PD and 



2 



PQ.QA=:QH.QG = QK and adding these two equations 



2 2 2 



PQ = PD + QK — Q. E. D. 



