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.