. 21 (B) of the mutual Relations of the Sides. (a) In a right-angled triangle, the square of the hypotenuse is equal to the squares of the two sides (b) In every triangle, the square of the side which is opposite to a given angle is greater or less than the squares of the sides containing that angle, by twice the rectangle contained by either of these sides, and that part of it which is intercepted between the perpendicular, let fall upon it from the opposite angle, and the acute angle; greater if the given angle is greater than a right angle, and less if it is less (c) Any angle of a triangle is equal to or greater or less than a right angle, according as the square of the opposite side is equal to or greater or less than the squares of the containing sides 22 cor. 21, 22 (d) In every triangle, if a perpendicular be drawn from the vertex to the base, the difference of the squares of the sides is equal to the difference of the squares of the segments of the base, i. e. the base is to the sum of the sides as the difference of the sides to the difference of the segments of the base or sum of the segments of the base produced 22 (e) In a right-angled triangle, if a perpendicular be drawn from the right angle to the hypotenuse, the square of the perpendicular is equal to the rectangle under the segments of the hypotenuse, and the square of either side is equal to the rectangle under the hypotenuse and the segment adjoining to it; i. e. the perpendicular is a mean proportional between the segments of the hypotenuse, and either side is a mean proportional between the hypotenuse and the segment adjoining to it cor. 21 and 61 (ƒ) In an isosceles triangle, if a straight line is drawn from the vertex to any point in the base, or in the base produced, the square of that straight line shall be less or greater than the square of the side by the rectangle under the segments of the base, or of the base produced 23 (g) In every triangle, the squares of the two sides are together double of the squares of half the base, and of the straight line which is drawn from the vertex to the middle point of the base 23 In every triangle, if the vertical or exterior vertical angle be bisected by a straight line which cuts the base, or the base produced, the base or base produced shall be divided in the ratio of the sides: also the square of the bisecting line shall be equal to the difference of the rectangles under the sides and the segments of the base, or base produced 70, 89 (i) In every triangle, if the base is equally produced both ways, so that the base produced is a third proportional to the base and sum of the sides, the sides of the triangle are to one another as the corresponding seg ments of the base produced sch. 64 (k) In every triangle, if the base is equally reduced both ways, so that the base reduced is a third proportional to the base and the difference of the sides, the sides of the triangle are to one another as the corresponding seg ments of the base reduced sch. 65 (7) In an isosceles triangle, if the equal angles are each of them double of the vertical angle, the sides and base are in extreme and mean ratio. See "Errata." Every triangle is a mean proportional between two rectangles, the sides of which are equal to the semiperimeter of the triangle and the excesses of the semiperimeter above the three sides. F The demonstration of this is briefly as follows:Let a circle be described within the triangle ABC, and a circle upon the other side of BC, the one touching B C, AC, and A B. in the points D, E, F, and the other BC and A C, A B produced in the points G, H, K; then the centres L, M of these circles lie in the straight line which bisects the angle at A, and if B L, BM be joined, the angle LBM will be a right angle, because it is half the sum of the angles A B C, C B K; also, be cause AK and AH are together equal to A B, BG and AC, CG, that is to the perimeter of the triangle ABC, each of them is equal to the semiperimeter. Again, because F B and B K (i. e. BD and BG) are together equal to E C and C H (i. e. to CD and CG), taking away GD from each, 2 BG is equal to 2C D, and consequently BG to CD; therefore F B is equal to CH, and BK to EC. AK therefore, is the semiperimeter, B K is the excess above the side A B, BF the excess above the side A C, and A F the excess above the side B C. K C D B G M H (c) Triangles which stand upon the same ther 17 (d) Triangles which have equal altitudes 1. The three angles of the one 60 ters, the isosceles has the greatest area 99 Problems relating to the Triangle. 1. Two sides and the included angle. 3. Two angles and the interjacent side. 4. Two angles and a side opposite to one of them. 5. The three sides 27 (b) To describe a triangle, when there are given the 1. Vertical angle, base, and sum (or 2. Vertical angle, base, and area. 4. Base, sum (or difference) of sides (c) To describe a triangle 1. Which shall be equal to a given rectilineal figure, and have a side and adjoining angle the same with a given side, and adjoining angle of the figure 28 2. Which shall be equiangular with a given triangle, and have a given perimeter [or area] 28 3. Which shall have for two of its sides the parts which are cut off by the third side from two straight lines given in position, and the third passing through a given point. 77 (d) To describe a right-angled triangle 74 proportionals, and the remaining Triplicate, one ratio said to be of another 34 angles of the same affection, or 61 one of them a right angle (1) Similar triangles are to one another in the duplicate ratio (or as the squares) of their homologous sides 67 (1) Of all triangles having the same two sides, that one has the greatest area, in which the angle contained by the two sides is a right angle. 103 (k) Of triangles which have equal bases, and equal areas, the isosceles has the least perimeter; and of triangles having equal bases and equal perime Undecagon. See "Hendecagon." Now the area of the triangle ABC is equal to LFX AK, for it is equal to half the rectangle under the radius L F of the inscribed circle, and the sum 2 A K of the three sides (I. 26. cor.); also BF BK is equal to LF x M K, because, LBM being a right angle, the right-angled triangles BFL, MKB are similar: but AKX AF: AKX FL :: AF FL, i. e. :: A K: KM, i. e. :: AKx FL KMX FL: therefore A Kx FL is a mean proportional between A K x AF and K M x FL or BF BK, that is (if a, b, c represent the three sides opposite to the angles A, B, C respectively and S the half of (a+b+c) the area of the triangle is a mean proportional between Sx (Sa) and (S - b) x (S-c). A TREATISE ON ALGEBRAICAL GEOMETRY. BY THE REV. S. W. WAUD, M.A. F. Ast. S. FELLOW AND TUTOR OF MAGDALENE COLLEGE, CAMBRIDGE, PUBLISHED UNDER THE SUPERINTENDENCE OF THE SOCIETY FOR THE DIFFUSION OF USEFUL KNOWLEDGE. LONDON: BALDWIN AND CRADOCK, PATERNOSTER-ROW. MDCCCXXXV. CONTENTS. PART I. APPLICATION OF ALGEBRA TO PLANE GEOMETRY. CHAPTER I. INTRODUCTION. Art. 1. Object of the Treatise 2. The method of expressing the length of a straight line by Algebra 3. The method of expressing the size of an area 4. The method of expressing the volume of a solid 5. General signification of an equation when referring to Geometry 6. Particular cases where the equation refers to areas and surfaces 7. Equations of the second and third order refer to some Geometrical Theorem 8. The solution of an equation leads either to numerical calculations, or to Geometrical Constructions 9, 10, 11. The Geometrical Construction of the quantities a+b, ab с Jab, Jab+cd, Noa2±b2, Noa2 + b2 + c2, No12, No} 12. Method of uniting the several parts of a construction in one figure 13. If any expression is not homogeneous with the linear unit, or is of the form a √ √2+b, &c., the numerical unit is understood, and must be expressed prior to construction. 7 CHAPTER II. DETERMINATE PROBLEMS. 14, Geometrical Problems may be divided into two classes, Determinate and Indeterminate an example of each. 15. Rules which are generally useful in working Problems 16. To describe a square in a given triangle. 17. In a right-angled triangle the lines drawn from the acute angles to the points of bisection of the opposite sides are given, to find the triangle 18. To divide a straight line, so that the rectangle contained by the two parts may be equal to a given square. Remarks on the double roots 9 |