Proposition 29. Theorem. 335. If two chords cut each other in a circle, the product of the segments of the one is equal to the product of the segments of the other. Hyp. Let the chords AB, CD cut at P. 336. DEF. When four quantities, such as the sides about two angles, are so related that a side of the first is to a side of the second as the remaining side of the second is to the remaining side of the first, the sides are said to be recipro cally proportional. Therefore 337. COR. 1. If two chords cut each other in a circle, their segments are reciprocally proportional. 338. COR. 2. If through a fixed point within a circle any number of chords are drawn, the products of their segments are all equal. EXERCISES. 1. If AP = 4, BP = 5, and CD = 12, find the lengths of CP and DP. = = 2. If AB 20, and CD 24, find the lengths of CP and DP when CD bisects AB. Proposition 30. Theorem. 339. If from a point without a circle a tangent and a secant be drawn, the tangent is a mean proportional between the whole secant and the external segment. Hyp. Let PC and PB be a tangent and a secant drawn from the point P to the O ABC. A Therefore, the square of the tangent is equal to the prod uct of the whole secant and the external segment. 341. COR. 2. Since PB x PA = PC, and PEX PD PC, = ... PB × PA = PE X PD. (340) (340) (Ax. 1) Therefore, if from a point without a circle two secants be drawn, the product of one secant and its external segment is equal to the product of the other and its external segment. 342. COR. 3. If from a point without a circle any number of secants are drawn, the products of the whole secants and their external segments are all equal. Proposition 31. Theorem. 343. If any angle of a triangle is bisected by a straight line which cuts the base, the product of the two sides is equal to the product of the segments of the base plus the square of the bisector. Hyp. In the A ABC let AD bisect the Z BAC. To prove AB × AC=BD × DC+AD2. Proof. Describe a O about the ▲ ABC, and produce AD to meet the Oce in E. Join CE. Then in the AS BAD, EAC, NOTE.-By means of this theorem we may compute the lengths of the bisectors of the angles of a triangle when the three sides are known. EXERCISE. If the vertical angle BAC be externally bisected by a straight line which meets the base in D, show that the product of AB, AC together with the square on AD is equal to the product of the segments of the base. Proposition 32. Theorem. 344. The product of two sides of a triangle is equal to the product of the diameter of the circumscribed circle by the perpendicular let fall upon the third side from the vertex of the opposite angle. Hyp. In the AABC let AD be to BC, and AE the diameter of the circumscribed O. To prove AB × AC = AE × AD. Proof. In the As ABD, AEC, B Ε NOTE.-By means of this theorem we may compute the radius of the circle circumscribed about a triangle when the three sides are known. EXERCISE. The product of the two diagonals of a quadrilateral inscribed in a circle is equal to the sum of the products of its opposite sides. PROBLEMS OF CONSTRUCTION. Proposition 33. Problem. 345. To divide a given straight line into parts proportional to given straight lines. Given, the line AB, and the lines P NM A M, N, P. Required, to divide AB into parts proportional to M, N, P. Cons. From A draw the indefinite straight line AH making any with AB. On AH take AC M, CDN, 7 G D Join EB, and through D, C draw DG, CF to EB. Then AB is divided at G and F into parts proportional If two st. lines are cut by any number of || s, the corresponding segments are proportional (300). 346. SCH. If it be required to divide a line AB into any number of equal parts, on AH, so that AC take the same number of equal parts CD = DE, and complete the con struction as before; then AF = FG EXERCISES. = GB. 1. Divide a straight line into five equal parts. 2. Give the construction for cutting off two-sevenths of a given straight line. |