91 



The Cayleyan Cubic. 

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



The Use of the Bicycle Wheel in Illustrating the Principles 



OF THE Gyroscope. 



By Chas. T. Knipp. 

 (PubHshed in the Physical Review, VoL XII, No. 1, January, 1901. 



The Cyclic Quadrilateral. 



By J. C. 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 nnd F'isher'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 ZPHG = ZGFI = ZQAG 

 .*. ZPAG is the supplement of ZPHG 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. 



