28 Mr. Davies's Symmetrical 



Then similar triangles give us 



VA . A« = GV . Frt, or VA . S = r^ . r„ 

 WA. AT = EW. FT, or WA . S = r, . r„ 

 GX . CU = EX . GU, or CX . S = r, . r,; 



and therefore by addition of equivalents 



r,.r, + r,.r, + r,.r, = [VA + AW + CX] . S = S=: 



or the sum of the rectangles, tinder the radii of external contact, 

 taken t-joo and t'jao, is equivalent to the square of the semiperi- 

 meter. A theorem due to Professor Lowry, of the R. M. 

 College.* 



Again, for the common expression for the area in terms of 

 the sides, we have 



BO . EW = eO . BW, by sim. trians. BOe, BWE; 



WA . AO =eO. EW, by sim. trians. EWA, KeO; 



BW = BW; hence by compounding, we obtain, 



BO . WA . AO . BW = eO" . BW '={eO. BW)- =area'. 



Also, by similar triangles, we have 



AP.PC = ES.fP, and 



\ A. AC = GV. Fa; whence, compounding 



AP.PC.VA. AC = ES.GV.Fa.<^P = area*. 



Or, the continued product of the four radii of' contact is equal 

 to the square of the area. — Hamilton's Analytical Geometry, 

 p. 45 : or Gent. Math. Comp. No. 22. 



To resume, — by sim. trians. 



VA . AP, or BO . AP = CV . fP = r . r„ 

 BP . PO, or CZ . BO = FP. eZ = /• . r,, 

 CX.cz, or AP. CZ = EX. eZ = ?.;,. 

 And by addition of equivalents, 



BO . AP + CZ . BO -F AP . CZ = r [r, + 7, + ; ,]. 

 Also, AP :PE:: Aa :rtF, or AP: r: : S : ?-3, 



CP:PE:: CA : VG, or CP:r::S:r„ 

 BO: OE: : BW: WE, or BO : r : : S : r,. 

 Or, compounding, 



AP.CP.BO:?-::S^::/-,.'i-r,. 



Compounding, again; these analogies in pairs, we shall obtain 



AP . CP : »- : : S^ : r, . r„ 



AP.BO:r^::S':/V'-3, 



CP . BO : r' : : S" : ;-, . > ,, ; and hence 



* See Levbouin's Reposilorv, No. 16. f). 5. 



^ AP. CP 



