S4 y^RlE DEMOnSTRAT GEOMETR. 



erit fiimma laterum .ABH-AC-i-BC:=a AR-I-2BP 

 ^- 2>C Q^, , ideoqiie habebitnr A R -i- B P 4- C Qjzz femi- 

 fummae latcrum — S. Erit ergo 

 AR -i- BC — S ideoque AR :=z AQ^— S - B C 

 B p -4- AC =: S ideoque BP-zBR = S-AC 

 C(l-\- AB =: S ideoque CQ_— C P = S - A B. 

 ;: Q; E. D. 



Theorema. 



*»S'5« §. S. Si ATt ,ante ex centro O cn-culi triangulo 



ABC infcripti in' fmgula latera demittantur perpendicula 

 O P , O Q^, O R , erit folidum fub partibus A R . B P . C Q^ 

 contentum aequale folido ex iemifumma laterum S et qua- 

 drato radii circuli inicripti OP confedo feu erit ; AR. 

 BP.CQ_=z:S.OP'. 



Demonftratio. 



Dudis ex centro circuli infcripti O ad fmgiilos an- 

 gulos redlis O A . O B . O C , ad earum aliquam C O fi 

 opus eft producftam ex altero reliquorum angulorum B 

 ducatur normalis BM , quae radio PO produfto occur- 

 rat in N. lam cum anguli A, B, C a recftis OA, OB, 

 OC bifu-iam fecentur , erit in triangulo BOC angulus 

 extremusBOM = ^BH-iC, hinc ob BOM-f-OBM=: 

 redo , erit |B-f-^C4-OBM= redo. Verum quia 

 A-f-B-f-Cr=2 red. erit quoque i B -}- ^ C -f- J A rr re- 

 ao, ideoque iB-f-iC-f-OBMrriB-f-^C-i-iA vn- 

 de fit OBM:=;iAi=:OAR. Quare cum in triangulis 

 redangulis BOM et AOR fit ang. OBMzzang.OAR 



ea 



