See N. Book VI. PRO P. B. THE OR. which likewise curs 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 bisected by the straight line AD; the rectangle BA, AC is equal to the rectangle BD, DC together with the square of AD. a. $. 4. Describe the circle * ACB about the triangle, and produce AD to the circumference in E, and join A is equal to the angle CAE, and the D triangles ABD, AEC are equiangu lar to one another. therefore as 6. 4. 6. BA to AD, so is EA to AC, and confequently the rectangle BA, AC d. 16. 6. is equal to the rectangle EA, AD, E that ise to the rectangle ED, DA together with the square of AD. but the rectangle ED, DA is £. 35. 3. equal to the rectangle f BD, DC. Therefore the rectangle BA, AC is equal to the rectangle BD, DC together with the square of AD. Wherefore if an angle, &c. Q. E. D. See N. IF from an angle of a triangle a straight line be drawn PROP. C. THEOR. 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 base BC; the rectangle BA, AC is equal to the rect. angle contained by AD and the diameter of the circle described about the triangle. Describe A a. s. 4 b. 31. 3. Describe a the circle ACB about Book VI. the triangle, and draw its diameter AE, and join EC. because the right angle BDA is equal to the angle ECA in a semicircle, and the angle B с D ABD to the angle AEC in the same segment"; the triangles ABD, c. 21. 3. AEC are equiangular, therefore as d. 4. 6. BA to AD, so is EA to AC, and consequently the rectangle BA, AC E is equal to the rectangle EA, AD. If therefore from an angle, &c. c. 16. 6. 0. E. D. PRO P. D. THE OR. TH HE rectangle contained by the diagonals of a quadri lateral inscribed in a circle, is equal to both the rectangles contained by its opposite sides. Let ABCD be any quadrilateral inscribed in a circle, and join AC, BD; the rectangle contained by AC, 5D 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, be- a. 21. 3. cause they are in the same segment; therefore the triangle ABD is equiangular to the triangle BCE. wherefore b as BC is to CE, so is BD В. b. 4. 6. to DA, and consequently the rectangle BC, 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 E ABE is equiangular to the triangle BCD. as therefore BA to AE, fo is BD to DC; wherefore the rectangle BA, DC is equal to the rectangle BD, AE, but the rectangle BC, AD has been Mewn 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 payaan Burtažis. THE c. 16. 6. A I. II. III. it makes right angles with every straight line meeting it in that IV. in one of the planes perpendicularly to the common section of V. tained by that straight line, and another drawn from the point in VI. two straight lines drawn from any the same point of their com- VII. Two See N. VII. Book XI. another, which two other planes have, when the said angles of VIII. IX. X. XI. to each, and which are contained by the same number of similar XII. XIII. that are opposite, are equal, similar, and parallel to one another ; XIV. XV. XVI. XVII. the center, and is terminated both ways by the superficies of the XVIII. triangle about one of the sides containing the right angle, which fide remains fixed. thc 1 Book XI. the Cone is called a right angled Cone; if it be less than the other side, an obtuse angled, and if greater, an acute angled Cone. XIX. The axis of a Cone is the fixed straight line about which the triangle revolves. XX. The base of a Cone is the circle described by that side containing the right angle, which revolves. XXI. A Cylinder is a solid figure described by the revolution of a right angled parallelogram about one of its sides which remains fixed. XXI. The axis of a cylinder is the fixed straight line about which the parallelogram revolves. XXIII. The bases of a cylinder are the circles described by the two revolving opposite sides of the parallelogram. XXIV. XXV. XXVI. XXVII. An Octahedron is a solid figure contained by eight equal and equilateral triangles. XXVIII. A Dodecahedron is a solid figure contained by twelve equal pentagons which are equilateral and equiangular. XXIX. DEF. A. gures whereof every opposite two are parallel. PROP. |