BOOK V. PROPORTION. SECTION I. SIMILAR FIGURES. DEF. 1. If the angles of a rectilineal figure, taken in order, are equal respectively to those of another, also taken in order, the figures are said to be equiangular. Each angle of the one is said to correspond to the angle equal to it in the other, and the sides joining the vertices of corresponding angles are termed corresponding sides. DEF. 2. Similar figures are such as are equiangular, and have their sides proportionals, the corresponding sides being homologous. DEF. 3. Similar figures are said to be similarly situated upon given straight lines, when those straight lines are corresponding sides of the figures. THEOR. I. Rectilineal figures that are similar to the same rectilineal figure are similar to one another. Let each of the rectilineal figures A and B be similar to the rectilineal figure C: then shall A and B be similar to one another. Because A is similar to C, therefore the angles of A taken in order are equal to the angles of C also taken in order, each to each; in like manner the angles of B taken in order are equal to the angles of C taken in order, each to each; but things that are equal to the same thing are equal to one another, therefore the angles of A taken in order are equal to the angles of B taken in order, each to each. Again, because A is similar to C, therefore the ratio of any two sides of A is equal to the ratio of the corresponding sides of C; in like manner the ratio of any two sides of B is equal to the ratio of the corresponding sides of C; but ratios that are equal to the same ratio are equal to one another, therefore the corresponding sides of A and B are proportional. Hence the figures A and B are similar to one another. Q.E.D. 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. Let ABC, DEF be two triangles having the angles at A, B respectively equal to the angles at D, E, and consequently (I. 25) the angle at C equal to the angle at F: then shall the triangles be similar, having AB to BC as DE to EF, BC to CA as EF to FD, and CA to AB as FD to DE. Apply the triangle ABC to the triangle DEF, so that B may fall on E and BA along ED, then BC will fall along EF, since the angle ABC is equal to the angle DEF. Let A and C fall at A' and C' in ED and EF, or in these sides produced. Then, because the angle EA'C' is equal to the angle EDF, therefore A'C' is parallel to DF, and therefore A'E : DE :: EC' : EF, that is, AB : DE :: BC : EF, and therefore, alternando, AB: BC:: DE: EF. In like manner it may be shown that BC: CA: EF: FD, and CA: AB:: FD: DE. *IV. 9, Cor. 2. Q.E.D. * Ex. I. If two straight lines be drawn from the same point A, one touching a circle in B, and the other cutting it in C and D, the triangles ABC, ABD are similar. Ex. 2. ABC is an acute-angled triangle, and AD, BE are the * The references are to Part I. of Book IV. perpendiculars from A and B on the opposite sides. Shew that the triangle CDE is similar to the triangle ABC. Ex. 3. With centre B and radius BC a circle is described, cutting AC, one of the equal sides AB, AC of an isosceles triangle, in D. Frove that BC is a mean proportional between AC, CD. Ex. 4. Two equal triangles are drawn upon the same base, and a straight line is drawn through them parallel to the base : shew that the parts of it intercepted between the sides of each triangle are equal to one another. 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. Let ABC, DEF be two triangles having the angle BAC equal to the angle EDF, and the side AB to the side AC as the side DE to the side DF: C E then shall the triangles be similar, having the angle ABC equal to the angle DEF, and the angle ACB to the angle DFE. Apply the triangle ABC to the triangle DEF, so that A may fall on D, and AB along DE then AC will fall along DF, since the angle BAC is equal to the angle EDF. Let B and C fall at B' and C' in DE and DF, or in these sides produced. Then, because AB : AC:: DE : DF, therefore DB': DE:: DC': DF, therefore B'C' is parallel to EF, Alternando. and therefore the angle DB'C' is equal to the angle DEF, and the angle DC'B' to the angle DFE, that is the angle ABC is equal to the angle DEF, and the angle ACB to the angle DFE. Thus the triangles ABC, DEF are equiangular, and therefore by Theor. 2 are similar. Q.E.D. Ex. 5. If the perpendicular drawn from the vertex of a triangle to the base fall within the triangle and is a mean proportional between the segments of the base, the triangle is rightangled. Ex. 6. If APB is a semicircle, of which AB is the diameter and C the centre, N a point on CB, and AB is produced to T, so that CT: AC :: AC: CN, and PT is the tangent drawn from T, then CNP is a right angle. 0: Ex. 7. If AB, CD, produced if necessary, intersect at E, and AE EC:: ED: EB, the points A, B, C, D are concyclic. Ex. 8. If two quadrilaterals have three angles of the one equal to three angles of the other, and the sides about one pair of angles proportional, the sides adjacent to equal angles being homologous terms of the proportion, then the quadrilaterals are similar. |