A history of Greek mathematics - Wilbourhall.org

one
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THE COLLECTION. BOOK VII 421
in K, L; and draw LAI parallel to AD meeting GE produced
in M.
HE.FG HE FG LH A£ LH
n
'
EG . EF~ EF EG ~ '
AF HK == EK
'
In exactly the same way, if BE produced meets LM in M'
we prove that
Therefore
(The proposition is
HB . CD LE
ED.BC EK
EE.FG EB.CD
EG.EF" ED.BC
proved for IfBCD and any other transversal
not passing through E 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
•
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 C
f
D be two points on AB, then
(a) if 2AB.CD = CB 2 , AD
Z ^AC 2 + DB 2 \
A
i—
C
1
D
1
B*
(b) if 2AC.BD = CD*, AB 2 = AD 2 + CB\