Ex. 249. If each side of an equilateral triangle be trisected, and the points of division nearest to each vertex be joined respectively, a hexagon is formed which is equiangular and equilateral. Ex. 250. Homologous medians of equal triangles are equal. Ex. 251. Hoinologous altitudes of equal triangles are equal. Ex. 252. Two isosceles triangles are equal if the vertical angle and the altitude upon an arm of the one are respectively equal to the vertical angle and the homologous altitude of the other. Ex. 253. If a median of a triangle is perpendicular to the base, the triangle is isosceles. Ex. 254. If in the pentagon ABCDE AB= BC, AE = CD, and ZA= 2 C, then BE = BD, and ZE= ZD. Ex. 255. If the opposite sides of a hexagon are parallel, and one pair of opposite sides are equal, all opposite sides are equal. A Ex. 256. If from the ends of the base, BC, of an isosceles triangle, ABC, equal parts, BD and CE, be laid off on one arm and the prolongation of the other, the line joining D and E is bisected by the base. Ex. 257. If through any point, D, on the bisector of an angle, A, a parallel be drawn to one of the sides, to meet the other side in B, then AB = BD. Ex. 258. Angles whose corresponding sides are perpendicular to each other, are either equal or supplementary. Ex. 259. Two equilateral triangles are equal if the altitude of one equals the altitude of the other. Ex. 260. Any straight line that passes through the midpoint of one of the diagonals of a parallelogram, bisects the parallelogram. Ex. 261. The number of all diagonals of a polygon of n sides is ^{" ). 2 Ex. 262. If a perpendicular be dropped from the vertex to the base of a triangle, each segment of the base will be smaller than the adjacent side of the triangle. Ex. 263. How many sides has a polygon, the sum of whose interior angles is equal to three times the sum of the angles of a hexagon ? Ex. 264. How many sides has an equiangular polygon, whose exterior angle equals the interior angle of an equilateral triangle ? Ex. 265. Prove the proposition of the sum of the interior angles of a polygon by joining any point within to the vertices of the polygon. Ex. 266. If the vertices of a triangle lie in the sides of another triangle, the perimeter of the first is less than the perimeter of the second. Ex. 267. The perpendiculars from two vertices of a triangle upon the • median drawn from the third vertex are equal. Ex. 268. The altitude upon the hypotenuse of a right triangle divides the figure into two triangles which are mutually equiangular. Ex. 269. If the upper base of an isosceles trapezoid is equal to the arms, the diagonals bisect the angles at the lower base. Ex. 270. The bisectors of the four angles of a parallelogram enclose a rectangle. Ex. 271. The lines joining the midpoints of the sides of a rectangle, taken in order, enclose a rhombus. Ex. 272. The lines joining the midpoints of the sides of a rhombus, taken in order, enclose a rectangle. Ex. 273. The lines joining the midpoints of the sides of any quadrilateral, taken in order, enclose a parallelogram. (Ex. 212.) Ex. 274. The lines joining the midpoints of opposite sides of any quadrilateral, bisect each other. (Ex. 273.) Ex. 275. If the vertical angle of an isosceles triangle is one-half of a base angle, a bisector of a base angle divides the figure into two isosceles triangles. * Ex. 276. If a line from one end of the base of an isosceles triangle to the opposite side divides the figure into two isosceles triangles, then the line is a bisector of the base triangle, and each base angle equals the double of the vertical angle. * Ex. 277. The midpoints of two opposite sides of a quadrilateral and the midpoints of the diagonals determine the vertices of a parallelogram. * Ex. 278. The lines joining the midpoints of the opposite sides of a quadrilateral and the line joining the midpoints of the diagonals meet in a point. (Exs. 277, 274.) Ex. 279. The bisectors of the exterior angles of a quadrilateral form a quadrilateral, the sum of whose opposite an les is equal to one straight angle. Ex. 280. A line from the vertex of an isosceles triangle to any point in the base is smaller than the arms. * Ex. 281. If from the midpoint, D, of the base of an isosceles triangle, ABC, a line be drawn to meet one arm in E, the prolongation of the other in F, then CE< CF. A D. Ex. 282. If in the triangle ABC AB > AC, and D is a point in the prolongation of BA, then DB>DC. * Ex. 283. Lines joining the midpoints of two opposite sides of a parallelogram to the ends of a diagonal trisect the other diagonal. 162. REMARK. – In order to prove that the sum of two lines, a and b, equals a third line, c, either (a) Construct the sum of a and b, and prove the line so obtained is equal to c, or B (6) Lay off a (or b) on c, and prove that the line representing the difference equals 6 (or a). Ex. 284. The sum of the perpendiculars dropped from any point in the base of an isosceles triangle to the legs, is equal to the altitude upon one of the arms. Ex. 285. The sum of the three perpendiculars dropped from any point of an equilateral triangle upon the sides is constant, and equal to the altitude of the triangle. (Ex. 284). Ex. 286. If the altitude BD of A ABC is intersected by another altitude in G, and E Hand HF are perpendicularbisectors, prove BG = 2 (HE). DE Ex. 287. The line joining the point of intersection of the altitudes of a triangle and the pointof intersection of the three perpendicular-bisectors, cuts off one-third of the corresponding median. (143 and Ex. 286.) Ex. 288. The points of intersection of the altitudes, medians, and perpendicular bisectors of a triangle lie in a straight line. Ex. 289. The diagonals of an isosceles trapezoid are equal. Ex. 290. If the opposite angles of a quadrilateral are equal, the figure is a parallelogram. * Ex. 291. If the diagonals of a trapezoid are equal, the trapezoid is isosceles. Ex. 292. The median drawn to any side of a triangle is less than half the sum of the other two sides. BOOK II THE CIRCLE. CONSTRUCTIONS -ooooo DEFINITIONS 163. A circumference is a curved line all points of which are equidistant from a point within called the center. A circle is a portion of a plane bounded by a circumference and is usually · read O ABC or OD. A radius is any straight line drawn from the center to the circumference, as DA. A diameter is a straight line passing through the center, and terminating in the circumference, as AE. 164. An arc is a part of the circumference. A semicircumference is half of the circumference. A minor arc is an arc less than a semicircumference; a major arc is an arc greater than a semicircumference. 165. A secant is a straight line intersecting the circumference in two points, as FG. A tangent is a straight line, which |