Ex. 1120. If two similar triangles, ABC and DEF, have their homologous sides parallel, the lines AD, BE, and CF, which join their homologous vertices, meet in a point. Ex. 1121. In an acute triangle side AB = 10, AC 7, and the projection of AC on AB is 3.4. Construct the triangle and compute the third side BC. Ex. 1122. Divide the circumference of a circle into three parts that shall be in the ratio of 1 to 2 to 3. Ex. 1123. The circles having two sides of a triangle as diameters intersect on the third side. Ex. 1124. Construct a circle equivalent to the sum of two given circles. Ex. 1125. Assuming that the areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles, prove that the bisector of an angle of a triangle divides the opposite side into segments proportional to the adjacent sides. Ex. 1126. In a circle of radius 5 a regular hexagon is inscribed. Determine (a) the area of one of the segments of the circle which are exterior to the hexagon; (b) the area of a triangle whose vertices are three successive vertices of the hexagon; (c) the area of the ring bounded by the circumference of the given circle and that of the circle inscribed in the hexagon. Ex. 1127. Find the locus of the extremities of tangents to a given circle, which have a given length. Ex. 1128. A ladder rests with one end against a vertical wall and the other end upon a horizontal floor. If the ladder falls by sliding along the floor, what is the locus of its middle point? Ex. 1129. An angle moves so that its magnitude remains constant and its sides pass through two fixed points. Find the locus of the vertex. Ex. 1130. The lines joining the feet of the altitudes of a triangle form a triangle whose angles are bisected by the altitudes. Ex. 1131. Construct a triangle, given the feet of the three altitudes. Ex. 1132. If the radius of a sector is 2, what is the area of a sector whose central angle is 152° ? Ex. 1133. The rectangle of two lines is a mean proportional between the squares on the lines. Ex. 1134. Show how to inscribe in a given circle a regular polygon similar to a given regular polygon. FORMULAS OF PLANE GEOMETRY 570. In addition to the notation given in § 270, the following will be used: APPENDIX TO PLANE GEOMETRY MAXIMA AND MINIMA 571. Def. Of all geometric magnitudes that satisfy given conditions, the greatest is called the maximum, and the least is called the minimum.* 572. Def. Isoperimetric figures are figures which have the same perimeter. PROPOSITION I. THEOREM 573. Of all triangles having two given sides, that in which these sides include a right angle is the maximum. Given & ABC and AEC, with AB and AC equal to AE and AC respectively. Let CAB be a rt. ≤ and ≤ CAE an oblique Z. To prove ▲ ABC > ▲ AEC. Draw the altitude EF. AABC and AEC have the same base, AC. Altitude AB > altitude EF. .. ▲ ABC > ▲ AEC. Q.E.D. 574. Cor. I. Conversely, if two sides are given, and if the triangle is a maximum, then the given sides include a right angle. HINT. Prove by reductio ad absurdum. *In later mathematics a somewhat broader use will be made of these terms. 575. Cor. II. Of all parallelograms having given sides, the one that is rectangular is a maximum, and conversely. Ex. 1135. Construct the maximum parallelogram having two lines of given lengths as diagonals. Ex. 1136. What is the minimum line from a given point to a given line? Ex. 1137. Of all triangles having the same base and altitude, that which is isosceles has the minimum perimeter. PROPOSITION II. THEOREM 576. Of all equivalent triangles having the same base, that which is isosceles has the least perimeter. Given equivalent ▲ ABC and AEC with the same base AC, and let AB = BC and AE EC. To prove AB + BC + CA < AE + EC + CA. Draw CFL AC and let CF meet the prolongation of AB at G. Draw EG and BE and prolong BE to meet GC at F. |