THEOREM XXII. Equiangular triangles ABC, DEF, are to one another in a duplicate proportion of their homologous or like sides; or as the squares AK, and DM of their homologous sides, fig. 40. Let the perpendiculars CG and FH be drawn, as well as diagonals BI and EL. The perpendiculars make the triangles ACG and DFH equiangular, and therefore similar (by theo. 16.) for because the angle CAG=FDH, aud the right angle AGC=DHF, the remaining angle ACG=DFH, (by cor. 2 theo. 5.) Therefore GC: FH:: (AC: DF: :) AB: DE, or which is the same thing, GC : AB :: FH: DE for FH multiplied by AB=AB multiplied by FH. By theo. 19. ABC: ABI:: (CG: AI or AB as before :: FH DE or DL : :) DFE :: DLE, therefore ABC ABI:: DFE: DLE or ABC: AK:: DFE : DM, for AK is double the triangle ABI, and DM THEOREM XXIII. Like polygons ABCDE, a bc de, are in a duplicate proportion to that of the sides AB, a b, which are between the equal angles A and B, and a and b, or as the squares of the sides AB, ab, fig. 41. By the hypothesis AB: ab:: BC: bc, and thereby also the angle B-b; therefore (by theo. 21.) BAC =b a c; and ACB-a c b; in like manner EAD=0 a d, and EDA=e d a. If therefore from the equal angles A, and a, we take the equal ones EAD+BAC e a d+b a c the remaining angle DAC-d a c, and if from the equal angles D and d, EDA-e d a be taken, we shall have ADC=a d ́c and in like manner if from C and c be taken BCA-b c a, we shall have ACD-a cd; and so the respective angles in every triangle, will be equal to those in the other. By theo. 22. ADC: a b c :: the square of AC to the square of ac, and also ADC : adc :: the square of AC, to the square of a c; therefore from equality of proportions ABC: a b c :: ADC: a dc, in like manner we may shew that ADC: a dc: EAD: ead: Therefore it will be as one antecedent is to one consequent, so are all the antecedents to all the consequents. 2 That is, ABC: a b c as the sum of the three triangles in the first polygon, is to the sum of those in the last. Or ABC will be to a bc, as polygon to polygon. The proportion of ABC to a b c (by the foregoing theo.) is as the square of AB is to the square of ab, but the proportion of polygon to polygon, is as ABC to a bc, as now shewn therefore the proportion of polygon to polygon is as the square of AB to the square of a b. THEOREM XXIV. Let DHB be a quadrant of a circle described by the radius CB; BH an arc of it, and DH its complement; HL or FC the sine, FH or CL its co-sine; BK its tangent DI its co-tangent; CK its secant, and CI its co-secant. fig. 8. 1. The co-sine of an arc is to the sine, as radius is to the tangent. 2. Radius is to the tangent of an arc, as the co 3. The sine of an arc is to its co-sine, as radius to its co-tangent. 4. Or radius is to the co-tangent of an arc, as its sine to its co-sine. 5. The co-tangent of an arc is to radius, as radius to the tangent. 6. The co-sine of an arc is to radius, as radius is to the secant. 7. The sine of an arc is to radius, as the tangent is to the secant. The triangles CLH and CBK, being similar, (by theo. 16.) 1. CL: LH:: CB: BK. 2. Or, CB: BK:: CL: LH. The triangles CFH and CDI, being similar. 3. CF (or LH): FH:: CD: DI. 4. CD:: DI:: CF (or LH): FH. The triangles CDI and CBK are similar: for the angle CIB=KCB, being alternate ones (by part 2. theo. 3.) the lines CB and DI being parallel: the angle CDI-CBK, being both right, and consequently the angle DCI-CKB, wherefore, 5. DI:CD::CB:BK. |