SCIENCE ABSOLUTE OP SPACE. 43 



let GS=HT, so, that the line ST meets the 

 ray BD prolonged in a certain K (which it is 

 evident can be made in a way like as in 4) ; 

 moreover take SR=SA, ROlllST, and O the 

 intersection of ray BK with RO; then AABR 

 = AABO (39), and so AABC>AABD (con- 

 tra hyp.). 



41. Equivalent triangles ABC, DEF 

 have the sums of their triangles equal. 



,L H F -^ or ^ et MN bisect 



M ^rV- s ^TTT" AC and BC > and P Q 



4-/1Q bisect DF and FE; 

 \l \ and take RS III MN, 

 ~s- E and TO III PQ ; the per- 

 pendicular AG to RS 

 will equal the perpendicular DH to TO, or one 

 for example DH will be the greater. 



In each case, the ODF, from center A, has 

 with line-ray GS some point K in common, 

 and (39) AABK=AABC=ADEF. But the 

 A AKB (by 40) has the same angle-sum as 

 ADFE, and (by 39) as AABC. Therefore 

 also the triangles ABC, DEF have each the 

 same angle-sum. 



In S the inverse of this theorem is true. 

 For take ABC, DEF two triangles having 

 equal angle-sums, and ABAL=ADEF; these 

 will have (by what precedes) equal angle-sums, 



