54 A SYLLABUS OF DEF. 2. S'milar figures are said to be similarly described upon given straight lines, when those straight lines are homologous sides of the figures. THEOR. I. Rectilineal figures that are similar to the same rectilineal figure are similar to one another. THEOR. 2. If two triangles have their angles respectively equal, they are similar, and those sides which are opposite to the equal angles are homologous. THEOR. 3. If two triangles have one angle of the one equal to one angle of the other and the sides about these angles proportional, they are similar, and those angles which are opposite to the homologous sides are equal. THEOR. 4. If two triangles have the sides taken in order about each of their angles proportional, they are similar, and those angles which are opposite to the homo logous sides are equal. THEOR. 5. If two triangles have one angle of the one equal to one angle of the other, and the sides about one other angle in each proportional, so that the sides opposite the equal angles are homologous, the triangles have their third angles either equal or supplementary, and in the former case the triangles are similar. Cor. Two such triangles are similar (1.) If the two angles given equal are right angles or obtuse angles. (2.) If the angles opposite to the other two homo logous sides are both acute or both obtuse, or if one of them is a right angle. (3.) If the side opposite the given angle in each triangle is not less than the other given side. THEOR. 6. If two similar rectilineal figures are placed so as to have their corresponding sides parallel, all the straight lines joining the angular points of the one to the corresponding angular points of the other are parallel or meet in a point; and the distances from that point along any straight line to the points where it meets corresponding sides of the figures are in the ratio of the corresponding sides of the figures. COR. Similar rectilineal figures may be divided into the same number of similar triangles. DEF. 3. The point determined as in Theor. 6 is called a centre of similarity of the two rectilineal figures. THEOR. 7. In a right-angled triangle if a perpendicular is drawn from the right angle to the hypotenuse it divides the triangle into two other triangles which are similar to the whole and to one another. COR. Each side of the triangle is a mean proportional between the hypotenuse and the adjacent segment of the hypotenuse; and the perpendicular is a mean pro portional between the segments of the hypotenuse. THEOR. 8. If from any angle of a triangle a straight line is drawn perpendicular to the base, the diameter of the circle circumscribing the triangle is a fourth proportional to the perpendicular and the sides of the triangle which contain that angle. THEOR. 9. If the interior or exterior vertical angle of a triangle is bisected by a straight line which also cuts the base, the base is divided internally or externally in the ratio of the sides of the triangle. And, conversely, if the base is divided internally or externally in the ratio of the sides of the triangle, the straight line 56 A SYLLABUS OF drawn from the point of division to the vertex bisects the interior or exterior vertical angle. SECTION 2. AREAS. THEOR. 10. If four straight lines are proportional the rectangle contained by the extremes is equal to the rectangle contained by the means; and, conversely, if the rectangle contained by the extremes is equal to the rectangle contained by the means the four straight lines are proportional. Cor. If three straight lines are proportional the rectangle contained by the extremes is equal to the square on the mean; and, conversely, if the rectangle contained by the extremes of three straight lines is equal to the square on the mean the lines are proportional. THEOR. II. If two chords of a circle intersect either within or without a circle the rectangle contained by the segments of the one is equal to the rectangle con tained by the segments of the other. OBS. This theorem has been proved in Book III, to which reference may be made for the corollaries. THEOR. 12. The rectangle contained by the diagonals of a quadrilateral is less than the sum of the rectangles contained by opposite sides unless a circle can be circumscribed about the quadrilateral, in which case it is equal to that sum. THEOR. 13. If two triangles or parallelograms have one angle of the one equal to one angle of the other, their areas have to one another the ratio compounded of the ratios of the including sides of the first to the in cluding sides of the second. COR. If two triangles or parallelograms have one angle of the one supplementary to one angle of the other, their areas have to one another the ratio compounded of the ratios of the including sides of the first to the including sides of the second. COR. The ratio compounded of two ratios between straight lines is the same as the ratio of the rectangle contained by the antecedents to the rectangle contained by the consequents. THEOR. 14. Triangles and parallelograms have to one another the ratio compounded of the ratios of their bases and of their altitudes. THEOR. 15. Similar triangles are to one another in the duplicate ratio of their homologous sides. THEOR. 16. The areas of similar rectilineal figures are to one another in the duplicate ratio of their homologous sides. COR. I. Similar rectilineal figures are to one another as the squares described on their homologous sides. COR. 2. If four straight lines are proportional and a pair of similar rectilineal figures are similarly described on the first and second, and also a pair on the third and fourth, these figures are proportional; and conversely, if a rectilineal figure on the first of four straight lines is to the similar and similarly described figure on the second as a rectilineal figure on the third is to the similar and similarly described figure on the fourth, the four straight lines are proportional. THEOR. 17. In any right-angled triangle, any rectilineal figure 58 A SYLLABUS OF PLANE GEOMETRY. described on the hypotenuse is equal to the sum of two similar and similarly described figures on the sides, SECTION 3. LOCI AND PROBLEMS. Loci. i. The locus of a point whose distances from two fixed straight lines are in a constant ratio is a pair of straight lines, passing through the point of intersection of the given lines, if they intersect, and parallel to them, if the lines are parallel. ii. The locus of a point whose distances from two fixed points are in a constant ratio (not one of equality) is a circle. PROB. I. To divide a straight line similarly to a given divided straight line. PROB. 2. To divide a straight line internally or externally in a given ratio. PROB. 3. From a given straight line to cut off any part required. PROB. 4. To find a fourth proportional to three given straight lines. PROB. 5. To find a mean proportional between two given straight lines. Prob. 6. On a straight line to describe a rectilineal figure similar to a given rectilineal figure. Prob. 7. To describe a rectilineal figure equal to one and similar to another given rectilineal figure. CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS, |