THE COLLECTION. BOOK VII 421 
in K, L; and draw LM parallel to AD meeting GH pro 
duced in M. 
HE.FG _ HE FG _ LH AF _ LH 
AF' HK 
Then 
EG :ef~ EF HG~ AF HK HK 
In exactly the same way, if DH produced meets LM in M' 
we prove that 
Therefore 
H E. CD 
HD. BG 
HE. FG 
LH 
HK' 
HE. CD 
HG. EF ~ HD . EG 
(The proposition is proved for HBGD and any other trans 
versal not passing through H by applying our proposition 
twice, as usual.) 
Props. 136, 142 are the reciprocal; Prop. 137 is a particular 
case in which one of the transversals is parallel to one of the 
straight lines, Prop. 140 a reciprocal of Prop. 137, Prop. 145 
another case of Prop. 129. 
The Lemmas 12, 13, 15, 17 (Props. 138, 139, 141, 143) are 
equivalent to the property of the hexagon inscribed in two 
straight lines, viz. that, if the vertices of a hexagon are 
situate, three and three, on two straight lines, the points of 
concourse of opposite sides are in a straight line; in Props. 
138, 141 the straight lines are parallel, in Props. 139, 143 not 
parallel. 
Lemmas 20, 21 (Props. 146, 147) prove that, when one angle 
of one triangle is equal or supplementary to one angle of 
another triangle, the areas of the triangles are in the ratios 
of the rectangles contained by the sides containing the equal 
or supplementary angles. 
The seven Lemmas 22, 23, 24, 25, 26, 27, 34 (Props. 148-53 
and 160) are propositions relating to the segments of a straight 
line on which two intermediate points are marked. Thus: 
Props. 148, 150. 
If G, D be two points on AE, then 
(a) if 2AB.CD = GB 2 , AD 2 = AG 2 + DB 2 ; 
A C D B 
(6) if 2AG.BD = CD 2 , AB 2 = AD 2 + GB 2 .