244 Opuscula 



5^-+0\"-f-0Z,"-i^20X . 0Y-f-20X. OZ+2OY . OZ = AE^. (4) 



Ejeclis hitic ex aequalionihus (2), (3), (4) quantilatibus OX 



-l-OY-f-OZ, et OX'-f-Oy-j-OZ'.deducemusreduciioneper- 

 acta 



;2 



oTV OY' V 0Z'^= 1^ - 2OX . OY-2OX . OZ - 2OY . OZ ; 



4 



seel ex aeqnationc (1) 



0X'-+-0~2 ^OZ'2 - OX . OY+OX . OZ+OY . OZ, 

 ergo 



OX . OY4-OX . OZ+OY . OZ —^——lOX . OY-aOX.OZ-aOY.OZ 



4 



ex qua facile infertur 



= O"XOZ-4-OY.0Z= 



4 ' 



-rT.2 

 OX.OY = OX.OZ-i-OY.OZ = 



ergo. 



THEOREMA I. 



4- Dactis e qiiovis punclo pcripheriae circuli perpendicu- 

 laribus ad latera triangnli aequilateri circumscripli, aggregaUnn 

 rectanguloriiQi, quae fieri possunt his perpendicularibus , blnis 



scd AB = BC=CD-DE = EF=ecc.Ergo 



A13CDEF=— ^(OG + OH + ON + OL + OM + ecc) 

 2 



lara vero punciuraO sit incenlro circuli polygono inscripti; normales OG, 

 OH, OKj OL, OM erunt aeqiiales illius radio. SI ergo vocetur r ra- 

 dius circuli, et n numerus lateruin polygoni,erit hujus area =: -J I- 



a 

 adeoque 



0G-f-OH-t-0K + OL4-0Mecc. = «r; 



ergo in quocumque polygono rcgulari ductis a quovis puncto ejus areae 

 normaliljiis ad laiera.erit liarum linearum aggrcgaium acqiiale lot ra- 

 diis circuli, qiiot erunt latera. Erit ergo in triangulo aequilatero 

 ABC(fig. I.) OX-+-OY + OZ=^5r.SedAE.rr5r; ergo 



OX+OY-f-OZ^AE. 



