But the rectangle under the fum and difference of the two fides of any triangle, is equal to the rectangle under the base and the difference of the fegments of the base (Cor. II. 16.); whence the rectangle of HC, CI is equal to the rectangle of AC, CD. And, in the fame manner, it may be proved, that the rectangle of HC, CI is alfo equal to the rectangle of CB, CF confequently the rectangle of AC, CD will be equal to the rectangle of CB, CF. Q.E.D. PROP. XXIX. THEOREM. If two right lines be drawn from any point without a circle, the one to cut it, and the other to touch it; the rectangle of the whole and external part of the one, will be equal to the fquare of the other. E Let CB, CA be any two right lines drawn from the point c, the one to cut the circle ADBG, and the other to touch it; then will the rectangle of CB, CF be equal to the fquare of CA. For, For, find E, the centre of the circle (III. 1.), and through the points E, C draw the line CEG; and join EA: Then, fince AC is a tangent to the circle, and EA is a line drawn from the centre to the point of contact, the angle CAE is a right angle (III. 12.) And, because EA is equal to EG, or ED, the line co will be equal to the fum of EA, EC, and CD will be equal to their difference. Since, therefore, the rectangle under the fum and difference of any two lines is equal to the difference of their fquares (II. 13.), the rectangle of CG, CD will be equal to the difference of the fquares of CE, EA. But the difference of the fquares of CE, EA is equal to the fquare of CA (Cor. II. 14.); therefore the rectangle of CG, CD will also be equal to the square of CA. And it has been fhewn, in the laft propofition, that the rectangle of CG, CD is equal to the rectangle of CB, CF; confequently the rectangle of CB, CF, will also be equal to the fquare of ca. Q.E.D. 1 PROP. XXX. THEOREM. If two right lines be drawn from a point without a circle, the one to cut it, and the other to meet it; and if the rectangle of the whole and external part of the one be equal to the fquare of the other, the latter will bea tangent to the circle. Let AB, AC be two right lines, drawn from any point A, without the circle CBD, the one to cut it, and the other to meet it: then, if the rectangle of AB, AE be equal to the fquare of AC, the line AC will be a tangent to the circle. For, let F be the centre; and from the point a draw AD to touch the circle at D (III. 10.); also join FD FA, FC. Then, fince AD is a tangent to the circle, and AEB cuts it, the rectangle of AB, AE is equal to the fquare of AD (III. 29.) But the rectangle of AB, AE is alfo equal to the fquare of ac (by Hyp.); whence the square of AC is equal to the fquare of AD, or AC equal to AD (II. 3.) And, because FC is equal to FD, AC to AD, and ar common to each of the triangles AFC, AFD, the angle ACF will also be equal to the angle ADF (I. 21.) But, fince AD touches the circle, and DF is a line drawn from the centre to the point of contact, the angle ADF is a right angle (III. 12.) The angle ACF, therefore, is also a right angle; and CF produced is a diameter of the circle. And fince a right line, drawn from the end of the diameter, at right angles to it, touches the circle (III. 10.), AC will be a tangent to the circle CBD, as was to be fhewn. BOOK IV. DEFINITIONS. 1. One rectilineal figure is faid to be infcribed in another, when all the angles of the one are in the fides of the other. 2. One rectilineal figure is faid to be defcribed about another, when all the fides of the one pass through the angular points of the other. 3. A rectilineal figure is faid to be inscribed in a circle, when all its angular points are in the circumference of the circle. 4. A rectilineal figure is faid to be described about a circle, when each fide of it touches the circumference of the circle. |