... DE = EB, DF= AF. Show now that the circle on AB will pass through D. 18. The circles described on the equal sides of an isosceles triangle as diameters will intersect at the middle point of the base. Join the middle points FD, ED. See the 18th Example, Book I., to prove that ED is parallel to AC, and DF to AB. ... AD is a parallelogram. Hence the circles described with E and F as centres, and radii EA, FA, will pass through D. 19. The greatest rectangle that can be inscribed in a circle is a square. B A E F D C We want to prove that ABC, half a square, is greater than AEC, half of any other rectangle. Draw BG parallel to AC, and let AE meet it in G. 20. An exterior angle of an inscribed quadrilateral is 30°. How many interior angles are known? C D B G E A 22. A tangent makes an angle of 60° with a chord. What portion of the circumference does the chord cut off? 23. If circles be described in and about an equilateral triangle, they have the same centre. 24. If equilateral triangles be described in and about a circle, the area of one will be four times the area of the other. Pass the sides of one through the angles of the other. 25. Two tangents from the same point outside a circle make equal angles with the chord joining the points of contact. 26. If the three points in which the inscribed circle meets the sides of a triangle be joined, the triangle formed will be acute-angled. 27. If tangents be drawn through the extremities of two diameters of a circle, a rhombus is formed. 28. Find the centre of a circle, cutting off equal chords from the sides of a triangle. It is the centre of the inscribed circle. Prove it. BOOK IV. RATIOS. DEFINITIONS. 1. Ratio is the relation with respect to magnitude, which one quantity bears to another of the same kind, and is the quotient arising from dividing the first by the second. The ratio of A to B is A ; or, as it is usually written, A: B; the ratio of 2 to 4, isor. 2. A proportion expresses the equality of two or more ratios. If the ratio of A to B be the same as the ratio of C to AC = or D, these quantities form a proportion; thus, A:B::C: D. Hence a proportion is an equation with both members fractional. 3. Ratio may be direct or inverse. If x and y be two variable quantities, such that, as x increases, y increases at the same rate (that is, as x doubles, y doubles, etc.), x and y have a direct ratio to each other. If, as x increases, y diminishes proportionally (that is, as x doubles, y becomes one-half its former value, etc.), x and y have an inverse ratio to each other. Since, in an inverse ratio, x is proportional to the reciprocal of y, an inverse ratio is frequently called a reciprocal ratio. 4. The first terms of the ratios (that is, the first and third terms of the proportion) are called antecedents, and the second and fourth, consequents. 5. The first and fourth terms are called extremes, and the second and third, means. 6. Quantities are in continued proportion when a consequent of one ratio is the same as the antecedent of the next. Thus, A:B::B:C::C:D, etc. 7. The second quantity is then said to be a mean proportional between the first and third; and the third is a third proportional to the first and second. 8. In a proportion the fourth term is said to be a fourth proportional to the other three taken in order. 9. Quantities are in proportion alternately, when antecedent is compared with antecedent, and consequent with consequent. If A : B :: C: D, then, alternately, A:C::B:D. 10. Quantities are in proportion inversely, when antecedent is made consequent, and consequent, antecedent. If A : B :: C: D, then, inversely, B:A::D :C. 11. Quantities are in proportion by composition, when the sum of antecedent and consequent is compared with either antecedent or consequent. If A:B::C: D, then, by composition, A+B: A or B :: C+D: Cor D. 12. Quantities are in proportion by division, when the difference of antecedent and consequent is compared with either antecedent or consequent. If A:B::C: D, then, by division, A-B: A or B :: C-D: Cor D. Proposition 1. Theorem. If four quantities be in proportion, the product of the extremes is equal to the product of the means. Corollary 1.-If three quantities be in continued proportion, the product of the extremes is equal to the square of the Corollary 2.-A mean proportional between two quantities is the square root of their product; for B = V A × C. Scholium.-When we speak of the product of two quantities, one of them at least must be a number. We cannot have the product of two lines, or of two surfaces, or of two solids, using these terms in their geometrical sense. When we, for convenience, use the expression product of two lines, we mean the number of units in the length of one, multiplied by the number of units of the same kind in the length of the other. Expressed in this way, lines become numerical quantities, and may be used as factors. If A and B be two lines, and C the common unit of measure, and if A contain C, m times, and B contain C, n times, then evidently A mC m n Hence, when we have the ratio of two lines, we may substitute the ratio of their numerical measures. Hence, the three succeeding propositions are true for lines as well as for numbers; and |