Opuscdla 255 



z=:rV2, et ideoy=ar'-+-r\^2, z'=ry^/'2.~u' ; hinc aequatio 

 xy=z'u' § 33, posilo loco y el z valoribus super expres- 

 sis, convertiiur in 



et ideo 



x'^+u'2=rVi(u'—x') (a) 



Ex acqualionibus autemy=x'-t-rV'2, z'=rV2 — it' deduci- 

 lur x'=j' — rVa, n'=r^2 — z'. Ponantnr hi valores x',u' in 

 aerjualionc x'y=u'z' , el habehimus, reduclione peracla, 



Additis acqualionibus (a) , (b) , erit earum summa 



Sedj' — a:'=rV2, 3'+ j^' — r/a ; ergo 



x'2-i-j- 2-)-z'2 1 m'2 — rv'a-^'V^ — 4''" 

 Ergo 



THEOREMA XV. 



36. Ductis e quovis puncto peripheiiae circuli reclis b'neis 

 perpendicularibus ad latera qnadrali inscripli, aggregaliira qua- 

 draiorum harnni perpendlcularium aequat quadialnm diametri. 



37. Ab aequalione J"' — x' = rv/2 delralialur aequatio ^' + «' 

 = r'^2\i erit 



J- — x' — s' — w'=::o 

 ideoquc 



y= x'-\- z'-t- w 



Ergo 



THEOREMA XVI. 



38. E quovis puncto peripheriae circuli ductis reclis lineis 

 perpendicularibus ad latera quadrali inscripli , earum quae 

 major est , aequat aggregatum trium caeterarum. 



og. Circulo FHG (iig. g) circumscriljatur pentagonum re- 

 gulare ABCDEj alque a puncto quovis O peripheriae ducan- 

 tur rectac OX = ar, OZ = z, 0U=«, OY=j, OT = /per- 

 pendiculares laleribus, vel eorum produclionibus. Ducatur et 

 linea AK normalis ad DC, el prolrahanlur latera AB, AE 



