THEOREMS. 114. The shortest line and the longest line which can be drawn from a given point to a given circumference pass through the centre. 115. The shortest chord that can be drawn through a given point within a given circle is to the diameter which passes through the point. 116. In the same circle, or in equal circles, if the sum of two arcs is greater than the circumference, the greater arc subtends the less chord, and conversely. 117. If ABC is an inscribed equilateral triangle, and Pis any point in the arc BC, then PA PB+ PC. HINT. On PA take PM equal to PB, and join BM. 118. In what kinds of parallelograms can a circle be inscribed? Prove your answer to be correct. 119. The radius of the circle inscribed in an equilateral triangle is equal to one-third of the altitude of the triangle. 120. A circle can be circumscribed about a rectangle. 121. A circle can be circumscribed about an isosceles trapezoid. 122. The tangents drawn through the vertices of an inscribed rectangle enclose a rhombus. 123. The diameter of the circle inscribed in a rt. ▲ is equal to the difference between the sum of the legs and the hypotenuse. 124. From a point A without a circle, a diameter AOB is drawn, and also a secant ACD, so that the part AC without the circle is equal to the radius. Prove that the DAB equals one-third the DOB. 125. All chords of a circle which touch an interior concentric circle are equal, and are bisected at the points of contact. 126. If two circles intersect, and a secant is drawn through each point of intersection, the chords which join the extremities of the secants are parallel. HINT. By drawing the common chord, two inscribed quadrilaterals are obtained. 127. If an equilateral triangle is inscribed in a circle, the distance of each side from the centre of the circle is equal to half the radius of the circle. 128. Through one of the points of intersection of two circles a diameter of each circle is drawn. Prove that the straight line joining the ends of the diameters passes through the other point of intersection. 129. A circle touches two sides of an angle BAC at B, C; through any point D in the arc BC a tangent is drawn, meeting AB at E and AC at F. Prove (i.) that the perimeter of the triangle AEF is constant for all positions of D in BC; (ii.) that the angle EOF is also constant. LOCI. 130. Find the locus of a point at three inches from a given point. 131. Find the locus of a point at a given distance from a given circumference. 132. Prove that the locus of the vertex of a right triangle, having a given hypotenuse as base, is the circumference described upon the given hypotenuse as diameter. 133. Prove that the locus of the vertex of a triangle, having a given base and a given angle at the vertex, is the arc which forms with the base a segment capable of containing the given angle. 134. Find the locus of the middle points of all chords of a given length that can be drawn in a given circle. 135. Find the locus of the middle points of all chords that can be drawn through a given point A in a given circumference. 136. Find the locus of the middle points of all secants that can be drawn from a given point A to a given circumference. 137. A straight line moves so that it remains parallel to a given line, and touches at one end a given circumference. Find the locus of the other end. 138. A straight rod moves so that its ends constantly touch two fixed rods which are to each other. Find the locus of its middle point. 139. In a given circle let AOB be a diameter, OC any radius, CD the perpendicular from C to AB. Upon OC take OM= CD. Find the locus of the point Mas OC turns about O. CONSTRUCTION OF POLYGONS. To construct an equilateral ▲, having given: 140. The perimeter. 141. The radius of the circumscribed circle. 143. The radius of the inscribed circle. 142. The altitude. To construct an isosceles triangle, having given : 144. The angle at the vertex and the base, 145. The angle at the vertex and the altitude. 146. The base and the radius of the circumscribed circle. 147. The base and the radius of the inscribed circle. 148. The perimeter and the alti tude. HINTS. Let ABC be the ▲ required, and EF the given perimeter. The altitude CD passes through the middle of EF, and the A AEC, BFC are isosceles. E To construct a right triangle, having given: 149. The hypotenuse and one leg. AD B 150. The hypotenuse and the altitude upon the hypotenuse. 151. One leg and the altitude upon the hypotenuse as base. 152. The median and the altitude drawn from the vertex of the rt. Z. 153. The radius of the inscribed circle and one leg. 154. The radius of the inscribed circle and an acute angle. 155. An acute angle and the sum of the legs. 156. An acute angle and the difference of the legs. To construct a triangle, having given: 157. The base, the altitude, and the Z at the vertex. 158. The base, the corresponding median, and the at the vertex. 159. The perimeter and the angles. 160. One side, an adjacent 4, and the sum of the other sides. 161. One side, an adjacent Z, and the difference of the other sides. 162. The sum of two sides and the angles. 163. One side, an adjacent 2, and radius of circumscribed O. 164. The angles and the radius of the circumscribed O. 165. The angles and the radius of the inscribed O. 166. An angle, the bisector, and the altitude drawn from the vertex. 167. Two sides and the median corresponding to the other side. 168. The three medians. To construct a square, having given: 169. The diagonal, 170. The sum of the diagonal and one side. To construct a rectangle, having given: 171. One side and the ▲ formed by the diagonals. 172. The perimeter and the diagonal. 173. The perimeter and the of the diagonals. 174. The difference of the two adjacent sides and the of the diagonals. To construct a rhombus, having given: 175. The two diagonals. 176. One side and the radius of the inscribed circle. 177. One angle and the radius of the inscribed circle. 178. One angle and one of the diagonals. To construct a rhomboid, having given : 179. One side and the two diagonals. 180. The diagonals and the formed by them. 181. One side, one 2, and one diagonal. 182. The base, the altitude, and one angle. To construct an isosceles trapezoid, having given: 183. The bases and one angle. 185. The bases and the diagonal. 184. The bases and the altitude. 186. The bases and the radius of the circumscribed circle. To construct a trapezoid, having given: 187. The four sides. 188. The two bases and the two diagonals. 189. The bases, one diagonal, and the formed by the diagonals. CONSTRUCTION OF CIRCLES. Find the locus of the centre of a circle: T 190. Which has a given radius r and passes through a given point P. 191. Which has a given radius r and touches a given straight line AB. 192. Which passes through two given points P and Q. 193. Which touches a given straight line AB at a given point P. 194. Which touches each of two given parallels. 195. Which touches each of two given intersecting lines. To construct a circle which has the radius r and which also: 196. Touches each of two intersecting lines AB and CD. 197. Touches a given line AB and a given circle K. 198. Passes through a given point P and touches a given line AB. 199. Passes through a given point P and touches a given circle K. To construct a circle which shall: 200. Touch two given parallels and pass through a given point P. 201. Touch three given lines two of which are parallel. 202. Touch a given line AB at P and pass through a given point Q. 203. Touch a given circle at P and pass through a given point Q. 204. Touch two given lines and touch one of them at a given point P. 205. Touch a given line and touch a given circle at a point P. 206. Touch a given line AB at P and also touch a given circle. 207. To inscribe a circle in a given sector. 208. To construct within a given circle three equal circles, so that each shall touch the other two and also the given circle. 209. To describe circles about the vertices of a given triangle as centres, so that each shall touch the two others. CONSTRUCTION OF STRAIGHT LINES. 210. To draw a common tangent to two given circles. 211. To bisect the angle formed by two lines, without producing the lines to their point of intersection. 212. To draw a line through a given point, so that it shall form with the sides of a given angle an isosceles triangle. 213. Given a point P between the sides of an angle BAC. To draw through Pa line terminated by the sides of the angle and bisected at P. 214. Given two points P, Q, and a line AB; to draw lines from P and Q which shall meet on AB and make equal angles with AB. HINT. Make use of the point which forms with P a pair of points symmetrical with respect to AB. 215. To find the shortest path from P to Q which shall touch a line AB. 216. To draw a tangent to a given circle, so that it shall be parallel to a given straight line. |