GEOMETRICAL PORISMS. 121 



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^iven by pofition. There may be found a ftraight line KD 

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

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

 cumference of the given circle, they fhall cut off from KG, 

 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 muft 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 KG, then BL muft be 

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

 angle GKF is equal to i\LB, that is, to GEF ; 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 j therefore the triangles 

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

 that is, by hypothefis, as a to /3 ; 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 u, is to /3, fo let AC be to BD. Through FI, D, G 

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

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

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

 circumference, to meet the- given lines in F, G ; GF fhall be to 

 DG as AC to BD, or as a to /3. The demonflration 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- 

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



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