PROPOSITION XIX. THEOREM. 300. In any triangle the product of two sides 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. A Let ABC be a triangle, and AD the perpendicular from A to BC. Describe the circumference A B C about the ▲ A B C. Draw the diameter A E, and draw E C. (two rt. A having an acute of the one equal to an acute of the other are PROPOSITION XX. THEOREM. 301. 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. Let ABCD be any quadrilateral inscribed in a circle, AC and BD its diagonals. § 280 .. A ABD and BCE, are similar, (two are similar when two of the one are equal respectively to two Whence of the other). A D, the medium side of the one, (the homologous sides of similar are proportional). $ 278 Cons. $203 § 280 (two are similar when two Whence of the one are equal respectively to two of the other). A B, the longest side of the one, : BD, the longest side of the other, (the homologous sides of similar are proportional). § 278 Adding these two equalities, or BD (AECE) = ABX CD + ADX BC, BDX AC = ABX CD + AD × BC. Q. E. D. Ex. If two circles are tangent internally, show that chords of the greater, drawn from the point of tangency, are divided proportionally by the circumference of the less. ON CONSTRUCTIONS. PROPOSITION XXI. PROBLEM. 302. To divide a given straight line into equal parts. A B Let A B be the given straight line. It is required to divide A B into equal parts. From A draw the indefinite line A 0. Take any convenient length, and apply it to A O as many times as the line A B is to be divided into parts. From the last point thus found on A O, as C, draw C B. Through the several points of division on A O draw lines Il to C B. These lines divide A B into equal parts, $274 (if a series of s intersecting any two straight lines, intercept equal parts on one of these lines, they intercept equal parts on the other also). Q. E. F. Ex. To draw a common tangent to two given circles. I. When the common tangent is exterior. II. When the common tangent is interior. 303. To divide a given straight line into parts proportional to any number of given lines. Let AB, m, n, and o be given straight lines. It is required to divide A B into parts proportional to the given lines m, n, and o. Draw FB. From E and C draw E K and CH to FB. (a line drawn through two sides of a to the third side divides those sides proportionally). .. AH: HK: KB:: AC CE : E F. Substitute m, n, and o for their equals A C, C E, and E F. Then AH: HK: KB:: m : n o. Q. E. F |