46 



INTRODUCTION. 



197. Definition. The right line passing throogh the 

 foci, and terminated by the curve, is the greater axis, and 

 the line bisecting it at right angles, the lesser axis. 



198. Theorem. A right line passing through any point 

 of an ellipsis, and making equal angles with the right lines 

 drawn to the foci, is a tangent to the ellipsis. 

 f r^::— _ Let AB make equal angles with 



BC and BD, then it will touch the 

 ellipsis in B. Let E be any other 

 point in AB. Produce DB, take 

 BFrrBC, and join CF, then AB 

 bisects the angle CBF, and CAB 

 is a right angle. Join EC, ED, 

 EF, GD,then EC=EF, and EC+ED=EF+ED, and is greater 

 than DF (79), or EC+BD, or GC+GD, therefore E is without the 

 ellipsis, and AB touches it, 



199. Theorem. The right lines, drawn from any point 

 of the ellipsis to the foci, are to each other as the square 

 of half the lesser axis to the square of the perpendicular 

 from either focus, on the tangent at that point. 



Let A and B be the foci, the 

 point of contact, and AD the per- 

 pendicular to the tangent CD, 

 draw BE and BF parallel to AD 

 and CD, produce AD each way, 

 and let it meet BF and BC in F 

 and G. Then since /_ACI>zz 

 BCEizDCG, CG=:AC ; and BG 

 =:AC+BC. And BFqrzBGq— FGq=BAq— FAq (118), therefore 

 BGq— BAq=FGq— FAq ; but (FG+FA). (FG— FA)=FGq-~ 

 FAq; and FG+FA=:2FD=:2BE, and FG— FAz=AG=:2AD ; also 

 BG=2BH, and BA=2BI, whence BGq>-BAqi=4HIq, therefore 



BKADizHIq, and BE=:I^, but BE : BC::AD : AC, and BErr 



,^BC Hlq BC Hlq 



AD. — zi — -i, oi- — zz. ' . 



AC AD AC ADfj 



