.(6) 



(12) 



Si niullipliceulur inter se istae asquationes, ita ut liabeatur productum 



sin P X sin S ,., . 



-:— 7- r— 7T-, prodibit 



sinQ X sinO ' '■ 



AO xAQ _ AOxBQ _ AQ X DO _ B QxDO _ sinPxsinS 

 CP X CS ■" CP X BS ~ CS X DP "~ BS X DP ~ sinQ x sinO ' 



Sed AB=AQ + BQ, AD = DO— AO, BC = BS— CS et CD=CP+PD 



erit ergo 



(AQ4-BQ) X (DO — AO) _ AO x AQ 

 (BS — CS) X (CP + PD) "~ CP X CS * 



Multiplicationibus eflfectis, prodit 



AQ X DO + BQ X DO — AQ X AO — BQ X AO _ AO X AQ 

 BS X CP — CS X CP + BS X DP — CS X DP ~ CP x CS' 

 Wulliplicando utrumque membrum per factum ex denominatoribus , 

 et reducendo , obtinetur 



AQ X DO X CP X CS — BQ X AO X CP X CS 

 = AO X AQ X BS X CP — AO X AQ X CS X DP. 

 Atqui ex relationibus (6) , consequitur 



AO X AQ X CP X BS = AO X BQ X CP X CS 

 AO X AQ X CS X DP = AQ X DO X CP X CS 

 AO X AQ X BS X DP = BQ X DO X CP X CS 

 ergo 2 AQ X DO X CP X CS = 2 AO x AQ x BS x CP. 



vel 



DOxCS = AOxBS, unde DO : AO = BS : CS. 



Aliunde ex figura deducuntur DO : AO = DP : CP et BQ : AQ = BS : CS j 

 ergo , ut jam vidimus, recta OS per diagonalium media transit. 



THEOREMA VII. 



In omni quadrilatero completo, tria diagonalium media in dii-ectuni. 

 sunt posita (iig. 17 ). 



Sujjponamus OPi" per dLagonalium AC et BD media traosire. 



Demonst. Ex prsecedenti Theoremate, erit 

 AD:BC = sinS: sinO AD x AB _ sin P X sinS 



AB: CD= sinP:sinQ' ^ BG X CD ~ siuQxsinO' 



\ 



