Book VI. PROB. B. THEOR. See Note. IF an angle of a triangle be bisected by a straight line, which likewise cuts the base; the rectangle con. tained by the sides of the triangle is equal to the rect. angle contained by the segments of the base, together with the square of the straight line bisecting the angle. b 21. 3. same Ć 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. Describe the circlea ACB about the triangle, and produce Α. segment; the triangles E AD, that is °, to the rectangle ED, DA, together with the square of AD : but the rectangle ED, DA is equal to the rectanglef 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. C 4.6. d 16. 6. e 3. 2. f 35. 3. PROP. C. THEOR. See Note. IF from any angle of a triangle a straight line be drawn perpendicular to the base; the rectangle con. tained 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 rectangle contained by AD and the diameter of the circle described about the triangle. Book VI. 2 5. 4. raigh с Describe a the circle ACB about ABD to the angle AEC in the saine c'21.3 CO recta d 4. 6. ther an THE rectangle contained by the diagonals of a See Note: quadilateral 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, 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 a 21. 3. b 4. 6.' с c 16.6. D angular to the triangle BCD: as therefore BA to AE, so is BD to A DC; wherefore the rectangle BA, DC is equal to the rectangle BD, AE: but the rectangle BC, AD has been shown equal to the rectangle BD, CE: therefore the whole rectangle AC, BD d is equal to the rect-d42 angle AB, DC, together with the rectangle AD, BC. Therefore the rectangle, &c. Q. E. D." • This is a Lemma of Cl. Ptolomæus, in page 9 of his peyaan pyro Tuis. THE ELEMENTS OF EUCLID. BOOK XI. DEFINITIONS. I. II. III. when it makes right angles with every straight line meeting IV. drawn in one of the planes perpendicularly to the common V. contained by that straight line, and another drawn from the VI. tained by two straight lines drawn from any the same point VII. one another, which two other planes have, when the said VIII. IX. A solid angle is that which is made by the meeting of more See Note. than two plane angles, which are not in the same plane, in one point. X. See Note. equal, each to each, and which are contained by the same XII. tuted betwixt one plane and one point above it in which they XIII. two that are opposite are equal, similar, and parallel to onc XIV. XV. XVI. XVII. through the centre, and is terminated both ways log the super- XVIII. angled triangle about one of the sides containing the right angle, which side remains fixed. If the fixed side be equal to the other side containing the right angle, the cone is called a right angled.cone; if it be i less than the other side, an obtuse angled, and if greater, an acute angled cone, 1 Book XI. XIX. The axis of a cone is the fixed straight line about which the triangle revolves. XX. XXI. right angled parallelogram about one of its sides, which re. XXII. 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. teral figures, whereof every opposite two are parallel. |