160. Through a given point to draw a line equally inclined to two given lines. 161. The sum of the perpendiculars from any point in the base of an isosceles triangle to the other sides is the same whatever point be taken. When the point is in the base produced, sum changes to difference. 162. To bisect a quadrilateral by a straight line drawn through a given angular point. 163. To construct a triangle having given (1) Two angles and a side opposite one of them; (2) Two sides and the contained angle; (3) Two sides and the angle opposite one of them (ambiguous case); (4) The base, the difference of the other two sides, and one base angle; (5) Two angles and the perimeter. 164. To find a point in a line such that lines drawn to it from two given points outside of it shall make equal angles with the given line. 165. To find a line the square on which will be equal to the difference of the squares on two given lines. 166. To construct an isosceles triangle having given the base and the vertical angle. 167. To construct a right-angled triangle having given one of the acute angles and the hypotenuse, or one of the acute angles and one side. 168. To construct a right-angled triangle having given the hypotenuse and the sum of the other two sides. 169. Find a square equal to the sum of three given squares. Hence show how to find a square equal to the sum of any number of given squares. 170. If two opposite sides of a parallelogram be bisected and the points of bisection be joined each to a corresponding opposite angle, the diagonal which the lines cross shall be trisected. 171. Find the angle contained by the two given sides of a triangle when the area is the greatest possible. 172. Every point in the line that bisects a given angle is equally distant from the arms of the angle. 173. Of all triangles having the same vertical angle and whose bases pass through a given point, the least is that whose base is bisected in that point. 174. To determine a point in a given straight line to which lines drawn from two given points at unequal distances from the given line may have the greatest difference possible. а 175. Through a given point to draw a straight line such that the segments intercepted by perpendiculars let fall from two given points shall be equal. 176. To trisect a right angle. 177. To trisect a straight line. 178. To divide a straight line into any number of equal parts. 179. The two sides of a triangle are together greater than twice the line drawn from the vertex to the base bisecting the vertical angle. 180. On the sides of any triangle ABC equilateral triangles are described all external; show that the lines, which join the angular points of triangle ABC with the vertices of the equilateral triangles on the sides opposite to them respectively, are all equal. 181. Construct a right-angled triangle having given one side, and the sum or difference of the hypotenuse and the other side. 182. If the exterior angles B and C at the base of triangle ABC be bisected by lines meeting at D, show that half angle A is the complement of angle D. 183. Through a given point between two lines draw a line to meet the given lines and be bisected at the given point. Is this always possible ? 184. From A and B perpendiculars are drawn to the opposite sides of triangle ABC meeting them at D and E; show that the lines joining D and E to the middle point of AB are equal; and that DE will be bisected by the perpendicular on it from the middle point of AB. 185. Construct a square having given the diagonal. 186. Given three straight lines meeting at a point, to draw a straight line cutting the three so that the segments of it intercepted by the lines shall be equal. 187. From a given point to draw three straight lines equal to three given lines so that their extremities shall be in a straight line and intercept equal portions of the straight line. Two of the given lines must be greater than twice the third. 188. Construct a triangle having given the base, the difference of the sides, and the difference of the angles at the base. 189. The hypotenuse of a right-angled triangle, together with the perpendicular on it from the right angle, are greater than the other two sides of the triangle. 190. Through two given points to draw straight lines forming with a given straight line an equilateral triangle. 191. Construct a triangle having the vertical angle four times each of the angles at the base. 192. To draw a straight line so that the part intercepted by the sides of a given triangle shall be equal to one given line and parallel to another. 193. What elements of a triangle must be given to determine it? Mention also the limitations to which the given elements in certain cases must be subject. 194. What are the limits to the magnitude of the two sides, containing the right angle, of a right-angled triangle? 195. If A be equidistant from B and from the straight line CD, any point in AB is nearer to B than to CD; but any point in BA, produced from B to A, is further from B than from CD. 196. To construct a right-angled triangle having given the hypotenuse and the difference of the other two sides. 197. If three lines be drawn making equal angles with the three sides of a triangle, towards the same parts, they will form a triangle equiangular with the given triangle. 198. In the base of a triangle to find a point from which lines drawn to each side of the triangle, parallel to the other side, shall be equal. 199. Iftwo triangles have two sides of the one equal respectively to two sides of the other, and the angle contained by the two sides of the one the supplement of the angle contained by the two sides equal to them of the other, the triangles shall be equal in area. 200. If two right-angled triangles have the hypotenuse and a side of the one equal to the hypotenuse and a side of the other, each to each, they shall be equal in every respect. 201. Íf a straight line be perpendicular to a finite straight line, the difference of the squares on the straight lines which join any point in the former with the extremities of the latter shall be constant. 202. And conversely, If a point be such that the difference of the squares on its distances from the extremities of a finite line is constant, the locus of that point shall be a straight line perpendicular to the finite line. 203. Of all equal triangles on the same base, the isosceles has the least perimeter. 204. Construct a triangle having each of the angles at the base one and a half times the vertical angle. 205. What are the agnitudes of the angles in a triangle which are in proportion to the numbers 1, 2, and 3? Also when in proportion to the numbers 1, 2, and 5 ? 206. Show that a triangle is right-angled whose sides are proportional to n2 – 1, 2n, and n2 + 1. 207. To find two points D and E in two given lines AB and AC, such that AD and DE together equal a given line and contain an angle equal to a given angle. 208. Construct a right-angled triangle having given the hypotenuse and the perpendicular on it from the right angle. 209. Construct a right-angled triangle having given the perimeter and one of the acute angles. 210. Of the three triangles which have for common vertex any point within a parallelogram, and for bases two adjacent sides of the parallelogram and the diagonal between them, the last is equal to the difference of the other two. If the point is not between the two adjacent sides, then difference changes to sum. 211. A median of a triangle bisects all lines parallel to the side of the triangle which it bisects and which are terminated by the other sides of the triangle. 212. In two given straight lines to find two points such that the three straight lines which join them with two given points without the lines respectively, and with each other, shall be the least possible. 213. The lines bisecting the three angles of a triangle are concurrent, i.e., pass through the same point. 214. The lines bisecting the vertical angle and the two exterior angles at the base of a triangle are concurrent. 215. The three medians of a triangle are concurrent. 216. The six triangles into which the whole triangle is divided by the medians are equal, and the point where the medians intersect is the point of trisection of each of them. 217. To construct a triangle having given the three medians. 218. The three perpendiculars from the vertices of a triangle on the opposite sides are concurrent. 219. The lines perpendicular to the sides of a triangle through their middle points are concurrent. 220. Find four points in a plane such that the line joining any two is at right angles to the line joining the other two. 221. To divide a straight line into two parts such that the sum of the squares on the parts shall be equal to the square on a given line which is less than the other given line. 222. Of three lines drawn from the vertex of a triangle to the base, the first bisecting the base, the second bisecting the vertical 'angle, and the third at right angles to the base; the second is always intermediate in position and magnitude to the other two; and the angle between second and third equals half the difference of the angles at the base. 223. The triangle formed by the three bisectors of the exterior angles of a triangle is such that the straight lines joining its vertices to the opposite vertices of the original triangle are perpendicular to the sides of the new triangle. 224. On the sides AB and AC of a triangle, parallelograms are described externally, the sides parallel to BA and CA are produced to meet in D, and DA is joined. Prove that a parallelogram on BC with the other side parallel and equal to DA is equal to the sum of the parallelograms on BA and CA. Hence prove I. 47, 225. If lines be drawn from a fixed point to the circunference of a circle, the locus of their middle points is a circle. 226. In the figure of I. 47 prove AD at right angles to KC, and A DBK = Á ABC = AECF. 227. If two triangles be on equal bases and between the same parallels, and a line be drawn parallel to the base intersecting their sides, the parts of this line intercepted in the two triangles shall be equal. 228. Through a given point between two straight lines not produced to meet, to draw a straight line such that if the three straight lines were produced they would all meet in the same point. 229. If any point be joined to the angles of a rectangle, the sum of the squares on the lines drawn to the one pair of opposite angles shall be equal to the sum of the squares on the lines drawn to the other pair of opposite angles. 230. If two straight lines which meet one another be cut by a third, and from the points of section two other straight lines be drawn, making with the first two the same angle which the cutting line makes with them respectively, the angle contained by the last two lines shall be double of the angle contained by the first two. Or, If a ray of light be twice reflected (angle of incidence angle of reflection) the angle between the original path of the ray and the path after second reflection is double the angle between the reflectors. F |