14 



Fig II. 



diagonal of FGHI also passes through R, and we have established the 

 following 



Theorem.— The diagonals of an inscribed quadrilateral meet in the 

 orthocenter of the triangle whose vertices are the center of the circle, 

 and the points where the opposite sides meet. 



(12) (See Fig. 1.) Since QK, QE, PC and PD are tangents to circle O, 

 the following theorem holds: If the diagonals of an inscribed quadri- 

 lateral meet in R, and its opposite sides meet in P and Q, and PR and 

 QR be drawn cutting the circle in E, K, C and D, then PD, PC, QK and 

 QE are tangent to the circle. 



(13) The diagonals of any quadrilateral inscribed in circle O. and 

 whose opposite sides meet in P and Q, will pass through R. 



(14) If any point I, in circle O be joined to P and Q and cutting 

 the circle in F and H, PF and QH will meet on the circumference as at G. 



