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PRO P. B. THE O R.
an angle of a triangle be bifected by a straight line,
which likewise cuts the base ; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square of the straight line bisecting the angle.
Let ABC be a triangle, and let the angle BAC be bifected by the straight line AD; the rectangle BA, AC is equal to the rectangle BD, DC together with the square of AD.
Describe the circle a ACB about the triangle, and produce AD to the circumference in E, and join EC. then because the angle BAD is equal to the angle CAE, and the
А. bo 21. 3. angle ABD to the angle b AEC, for
they are in the same segment; the
B €. 4. 6. BA to AD, so is © EA to AC, and
consequently the rectangle BA, AC
d. 16. 6.
c. 3. 2.
f. 35. 3.
PROP. C. THEO R.
drawn perpendicular to the base ; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle.
Let ABC be a triangle, and AD the perpendicular from the angle A to the bafc BCthe rectangle BA, AC is equal to the rectangle contained by AD and the diameter of the circle defcribed about the triangle.
b. 31. 3•
Describe the circle ACB about
А. the triangle, and draw its diameter
a. 5. 4. AE, and join EC. because the right angle BDA is equal o to the angle
C ECA ¡a a semicircle, and the angle
D ABD to the angle AEC in the same fegmento; tbe triangles ABD, AEC
d. 4. 6. are equiangular. therefore as - BA to AD, fo is EA to AC, and conse E quently the rectangle BA, AC is equal to the rectangle EA, AD. If therefore from an angle, &c. c. 16. 6. O. E. D.
C. 21. 3•
PROP. D. THEOR.
drilateral inscribed in a circle, is equal to both the rectangles contained by its opposite sides.
Let ABCD be any quadrilateral infcribed in a circle, and join AC, BD; the rectangle contained by AC, BD is equal to the two rectangles contained by AB, CD and by AD, BC *.
Make the angle ABE equal to the angle DBC; add to each of these the common angle EBD, then the angle ABD is equal to the angle EBC. and the angle BDA is equal“ to the angle BCE, because a. 21. 3. they are in the same segment; therefore the triangle ABD is equiangular to the triangle BCE. whereforebas BCis to CE, fo is BD to DA,
b. 4. 6. and consequently the rectangle BC,
с AD is equal to the rectangle BD, CE. again, because the angle ABE is equal to the angle DBC, and the angle *BAE to the angle BDC, the triangle ABE is equiangular to the triangle BCD. as therefore BA to AE, fo is
A BD to DC; wherefore the rectangle. BA, DC is equal to the rectangle BD, AE. but the rectangle BC, AD has been shewn equal to the rectangle BD, CE; therefore the whole rectangle AC, BD is equal to the rectangle AB, DC together with the rectangle AD, BC. Therefore the rectangle, &c. Q.E.D.
This is a Lemma of Cl. Ptolomacus in page 9. of his siyáan ouvražis.
C. 16. 6.
A straight line is perpendicular, or at right angles, to a plane,
when it makes right angles with every straight line meeting it
in one of the planes perpendicularly to the common section of
tained by that straight line, and another drawn from the point
by two straight lines drawn from any the fame point of their
Book XI. Two planes are said to have the same, or a like inclination to one
another, which two other planes have, when the said angles of inclination are equal to one another.
VIII. Parallel planes are such which do not meet one another tho' produced.
IX. A solid angle is that which is made by the meeting of more than Se N.
two plane angles, which are not in the same plane, in one point.
X. · The tenth Definition is omitted for reasons given in the Notes.' See N.
XI. Similar solid figures are such as have all their solid angles equal, See N.
each to each, and which are contained by the same number of similar planes.
that are opposite, are equal, similar, and parallel to one another;
XIV. A Sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains unmoved.
the center, and is terminated both ways by the superficies of
angled triangle about one of the sides containing the right
Book XI. If the fixed fide be equal to the other side containing the right
angle, the Cone is called a right angled Cone ; if it be less
angled parallelogram about one of its fides which remains fix-
figures whereof every opposite two are parallel.