GEOMETRICAL FORISMS. laj 



given by pofition. There may be found a flraight line KD 

 given by pofition, and alfo a given point D in that line, 

 fuch, that if AE, BE be infledled to any point in the cir- 

 cumference of the given circle, they fliall cut off from KC, 

 KD, fegments FC, GD, adjacent to the given points, and 

 having to each the given ratio of a. to /3. 



Suppose the line KD, and the point D to be found. If AH, 

 BH be infledled to the circle, fo that AH may pafs through C, 

 then BH muil pafs through D, the point which may be found, 

 otherwife the propofition w^ould not be univerfally true. Now, 

 C being given, the point H, and the line BH, will be given by 

 pofition. Let AL be drawn parallel to KC, then BL mufl be 

 parallel to KD, the line to be found ; hence it appears, that the 

 angle GKF is equal to ALB, that is, to GEE ; therefore the 

 points E, K, G, F are in a circle, and the angle DGB is equal to 

 CFA ; now DBG is equal to CAF ; therefore the triangles 

 DBG, CAF are equiangular, and AC is to BD as CF to DG, 

 that is, by hypothefis, as a to /S ; now AC is given, and BH is 

 given by pofition, therefore the point D is given, but BDG is 

 equal to the given angle ACF, therefore DG is given by pofi- 

 tion. 



Construction. Join AC, meeting the circle in H. Join 

 BH, and, as a is to /3, fo let AC be to BD. Through H, D, C 

 defcribe a circle to meet FC in K. Join DK ; then D is the 

 given point, and DK is die line given by pofition, which are 

 to be fouijd ; that is, if AE, BE be infledled to any point in the 

 circumference, to meet the- given lines in F, G ; CF fliall be to 

 DG as AC to BD, or as « to jS. The demonfl:ration is eafily de- 

 rived from the analyfis. 



The foregoing propofitions, in one point of view, may be 

 confidered as exhibiting innumerable folutions of certain geo- 

 mietrical problems of the indeterminate kind, to each of which, 



?2 if 



