424 
EUCLID 
given, the triangle is given in species. Proposition 52 : If a 
(rectilineal) figure given in species be described on a straight 
line given in magnitude, the figure is given in magnitude. 
Proposition 66: If a triangle have one angle given, the rect 
angle contained by the sides including the angle has to the 
(area of the) triangle a given ratio. Proposition 80 : If a 
triangle have one angle given, and if the rectangle contained 
by the sides including the given angle have to the square on 
the third side a given ratio, the triangle is given in species. 
Proposition 93 is interesting: If in a circle given in magni 
tude a straight line be drawn cutting off a segment containing 
a given angle, and if this angle be bisected (by a straight line 
cutting the base of the segment and the circumference beyond 
it), the sum of the sides including the given angle will have a 
given ratio to the chord bisecting the angle, and the rectangle 
contained by the sum of the said sides and the portion of the 
bisector cut off (outside the segment) towards the circum 
ference will also be given. 
Euclid’s proof is as follows. In the circle ABC let the 
chord BC cut off,a segment containing a given angle BAC, 
and let the angle be bisected by AE meeting BC in D. 
Join BE. Then, since the circle is given in magnitude, and 
BC cuts off'a segment containing a given 
angle, BC is given (Prop. 87). 
Similarly BE is given ; therefore the 
ratio BG-.BE is given. (It is easy to 
see that the ratio BG-.BE is equal to 
2 cos f A.) 
Now, since the angle BAC is bisected, 
* BA:AC=BD: DC. 
It follows that {BA + AG): {BD + DC) = AG;DC. 
But the triangles ABE, ADC are similar; 
therefore AE -. BE = AG: DC 
= {BA + AC): BC, from above. 
Therefore {BA + AG): AE — BC: BE, which is a given 
ratio.