33 The Three Sections, Tangencies, 



AOH evidently becomes incleterminate when O and A coin- 

 cide, and that QOAT is parallel to XN. In this state of the 

 data all points in MINI are answerable points C, and the 

 })roblem is said to be " porismatic/'' It is evident that UA 

 is parallel to INIM when QOA is to NN^ and that m.n (in this 

 porismatic case) is equal QA.UA. 



THE DETERMINATE SECTION. 



Given two pair of points P,S^ and Q,R, in a straight line MN ; 

 to find a point C in the line such that PC . SC ; QC . IIC 

 ; / ; k . fin which the sign of ^ is hiown, as loell as its 

 magnitude) . 



ANALYSIS. 



Suppose we assume a point A, and that we draw PG and 

 Q,G making the angles PG right to Q, and QG right to P 

 respectively equal to RA right to S, and SA right to R ; and 

 that E and F are the points in which AC cuts QG and PG. 



The triangles CPE_, CQF, are similar to CRA, CSA, and we 

 evidently have PE . SA : QF . RA : : PC . SC : QC . 

 RC : : / : /f ; and therefore PE has to QF the known ratio 

 of RAi to SA.^. Hence (see Porism 4 in Transactions 

 for 1859), the circle EFG passes through a known point O in 

 the circumference of circle PGQ^ which is such that PO : 

 QO : : PE : QF. 



Again, the angle EF or EC right to O = GF or GQ right 

 to O = PQ or PC right to O ; therefore a circle can pass 

 through EPC and O ; and hence, as AR is parallel to PE, if 

 H be the point of intersection of PO and AR, it follows that 

 a circle can pass through OHA and C ; but O, li, and A are 

 known points ; therefore the circle OHA is known, and also 

 the point C in which it cuts MN. 



COMPOSITION. 



Assume a point A (not in the given line) ; draAV PG and 

 QG making the angles PG and QG right to M equal re- 



