a (i. 47) to the squares of d f, fe, because efd is a right angle: and the square of ec is equal to the squares of cf, fe; therefore the rectangle ad, dc, together with the square of e c, is equal to the square of ed: and ce is equal to eb; therefore the rectangle a d. dc, together with the square of eb, is equal to the square of ed; but the squares of e b, bd are equal to the square (i. 47) of ed, because ebd is a right angle; therefore the rectangle a d, dc, together with the square of eb, is equal to the squares of eb, bd. Take away the f common square of eb; therefore the remaining rectangle a d, dc is equal to the square of d b. Wherefore, if from any point, &e. Q. E. D. Cor. If from any point without a circle, there be drawn two straight lines cutting it, as a b, a c, the rectangles contained by the whole lines, and the 'c parts of them without the circle, are equal to one another, viz. the rectangle ba, a e, to the rectangle ca, af: for each of them is equal to the square of the straight line a d, which touches the circle. PROPOSITION XXXVII.-THEOREM. If from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it ; and if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square of the line which meets it, the line which meets shall touch the circle. LET any point d be taken without the circle a bc, and from it let two straight lines dca and db be drawn, of which dca cuts the circle, and db meets it ; if the rectangle a d, dc be equal to the square of db; db touches the circle. Draw (iii. 17) the straight line de touching the circle a b c, find its centre f, and join fe, fb, fd; then fed is a right (iii. 18) angle : and because de touches the circle a b c, and dca cuts it, the rectangle a d, dc is equal (iii. 36) to the square of de: but the d rectangle a d, do is, by hypothesis, equal to the square of d b: therefore the square of d e is equal to the square of db; and the straight line de equal to the straight line db; and fe is equal to fb, wherefore de ef are equal to db, bf; and b. the base fd is common to the two triangles def, le dbf; therefore the angle def is equal (i. 8) to the angle dbf; but def is a right angle, theref fore also d bf is a right angle : and fb, if produced, is a diameter; and the straight line which is drawn at right angles to a diameter, from the extremity of it, touches (iii. 16) the circle : therefore bd touches the circle a b c. Wherefore, if from a point, &c. Q. E. D. 73 EXERCISES ON BOOK III. SECT. I.- PROBLEMS. 1. Given a circle, a square, and a point without the circle, to draw from the point a straight line cutting the circle, such that the rectangle contained by the part of the line within, and the part without the circle, shall be equal to the given square. 2. Given a circle and a point within it which is not the centre, to draw through that point a chord which will be bisected in the point. 3. Given a circle with its diameter produced to a given point, to find in the part produced a point, such that if a tangent be drawn from it to the circle, the tangent shall be equal to the part of the produced diameter lying between the point and the given point. 4. Given a circle with the diameter produced and a straight line, to draw a tangent from the diameter produced to the circle, equal to the given straight line. 5. Given two circles, to draw a straight line which shall touch them. 6. Given the vertical angle, the altitude and the sum of the three sides of a triangle, to construct the triangle. 7. Given the vertical angle, the altitude and the base of a triangle, to construct the triangle. 8. Given a segment of a circle and a straight line, to describe on the line another segment of a circle which shall be similar to the given segment. 9. Given a point, a straight line, and a point in the line, to describe a circle which shall pass through the first point, and touch the straight line in the other point. SECT. II.-THEOREMS. 10. The line drawn from the centre of a circle perpendicular to any chord is also perpendicular to all chords parallel to the former. 11. If two parallel chords in a circle are bisected, the bisecting line is also perpendicular to them. 12. If from any two points in the circumference of the greater of two concentric circles, chords are drawn so as to touch the circumference of the lesser circle, these chords shall be equal. 13. If two parallel chords are drawn in a circle, the arcs intercepted between their extremities shall be equal. 14. If any point be taken without a circle, there cannot be drawn from that point more than two equal straight lines to touch the circumference of the circle. 15. If an equilateral triangle be inscribed in a circle, and from a point in the circumference straight lines be drawn to the three angles of the triangle, the greatest of these lines is equal to the sum of the other two. 16. If a quadrilateral be described about a circle, the angles subtended by any two opposite sides of the figure at the centre of the circle shall be equal to two right angles. 17. If two circles cut each other, the straight line wbich joins their centres will bisect the straight line which joins the points of intersection. BOOK IV. DEFINITIONS. I. A RECTILINEAL figure is said to be inscribed in another rectilineal figure, when all the angles of the inscribed figure are upon the sides of the figure in which it is inscribed, each upon each. II. In like manner a figure is said to be described about another figure, when all the sides of the circumscribed figure pass through the angular points of the figure about which it is described, each through each. Definitions I. and II. III. A rectilineal figure is said to be inscribed in a circle, when all the angles of the inscribed figure are upon the circumference of the circle. Definition III. IV. A rectilineal figure is said to be described about a circle, when each side of the circumscribed figure touches the circumference of the circle. V. In like manner, a circle is said to be inscribed in a rectilineal figure, when the circumference of the circle touches each side of the figure. VI. A circle is said to be described about a rectilineal figure, when the circumference of the circle passes through all the angular points of the figure about which it is described. VII. A straight line is said to be placed in a circle, when the extremities of it are in the circumference of the circle. PROPOSITION 1.- PROBLEM. In a given circle to place a straight line equal to a given straight line not greater than the diameter of the circle. LET a b c be the given circle, and d the given straight line, not greater than the diameter of the circle. Draw b c the diameter of the circle a b c; then, if b c is equal to d, the thing required is done, for in the circle a a b c a straight line bc is placed equal to d; but, if it is not, bc is greater than d; make ce equal (i. 3) to d, and from the b centre c, at the distance ce, describe the circle a ef, and join ca: therefore, because c is the centre of the circle a ef, ca is equal d to ce: but d is equal to ce; therefore d is equal to ca. Wherefore in the circle a b c, a straight line is placed equal to the given straight line d, which is not greater than the diameter of the circle. Which was to be done. PROPOSITION II.-PROBLEM. In a given circle to inscribe a triangle equiangular to a given triangle. LET a b c be the given circle, and def the given triangle ; it is required to inscribe in the circle a b c a trid angle equiangular to the triangle ha def Draw (iii. 17) the straight line gah touching the circle in the point a, and at the point a, in the f straight line a h, make (i. 23) the bo angle hac equal to the angle def; and at the point a, in the straight line a g, make the angle g a b equal e |