I. 46 to 48. 125. On the sides AC, BC of a triangle ABC, squares ACDE, BCFH are described: shew that the straight lines AF and BD are equal. 126. The square on the side subtending an acute angle of a triangle is less than the squares on the sides containing the acute angle. 127. The square on the side subtending an obtuse angle of a triangle is greater than the squares on the sides containing the obtuse angle. 128. If the square on one side of a triangle be less than the squares on the other two sides, the angle contained by these sides is an acute angle; if greater, an obtuse angle. 129. A straight line is drawn parallel to the hypotenuse of a right-angled triangle, and each of the acute angles is joined with the points where this straight line intersects the sides respectively opposite to them: shew that the squares on the joining straight lines are together equal to the square on the hypotenuse and the square on the straight line drawn parallel to it. 130. If any point P be joined to A, B, C, D, the angular points of a rectangle, the squares on PA and PC are together equal to the squares on PB and PD. 131. In a right-angled triangle if the square on one of the sides containing the right angle be three times the square on the other, and from the right angle two straight lines be drawn, one to bisect the opposite side, and the other perpendicular to that side, these straight lines divide the right angle into three equal parts. 132. If ABC be a triangle whose angle A is a right angle, and BE, CF be drawn bisecting the opposite sides respectively, shew that four times the sum of the squares on BE and CF is equal to five times the square on BC. 133. On the hypotenuse BC, and the sides CA, AB of a right-angled triangle ABC, squares BDEC, AF, and AG are described: shew that the squares on DG and EF are together equal to five times the square on BC. II. 1 to 11. 134. A straight line is divided into two parts; shew that if twice the rectangle of the parts is equal to the sum of the squares described on the parts, the straight line is bisected. 135. Divide a given straight line into two parts such that the rectangle contained by them shall be the greatest possible. 136. Construct a rectangle equal to the difference of two given squares. 137. Divide a given straight line into two parts such that the sum of the squares on the two parts may be the least possible. 138. Shew that the square on the sum of two straight lines together with the square on their difference is double the squares on the two straight lines. 139. Divide a given straight line into two parts such that the sum of their squares shall be equal to a given square. 140. Divide a given straight line into two parts such that the square on one of them may be double the square on the other. 141. In the figure of II. 11 if CH be produced to meet BF at L, shew that CL is at right angles to BF. 142. In the figure of II. 11 if BE and CH meet at 0, shew that AO is at right angles to CH. 143. Shew that in a straight line divided as in II. 11 the rectangle contained by the sum and difference of the parts is equal to the rectangle contained by the parts. II. 12 to 14. 144. The square on the base of an isosceles triangle is equal to twice the rectangle contained by either side and by the straight line intercepted between the perpendicular let fall on it from the opposite angle and the extremity of the base. 145. In any triangle the sum of the squares on the sides is equal to twice the square on half the base together with twice the square on the straight line drawn from the vertex to the middle point of the base. 146. ABC is a triangle having the sides AB and AC equal; if AB is produced beyond the base to D so that BD is equal to AB, shew that the square on CD is equal to the square on AB, together with twice the square on BC. 147. The sum of the squares on the sides of a parallelogram is equal to the sun of the squares on the diagonals. 148. The base of a triangle is given and is bisected by the centre of a given circle: if the vertex be at any point of the circumference, shew that the sum of the squares on the two sides of the triangle is invariable. 149. In any quadrilateral the squares on the diagonals are together equal to twice the sum of the squares on the straight lines joining the middle points of opposite sides. 150. If a circle be described round the point of intersection of the diameters of a parallelogram as a centre, shew that the sum of the squares on the straight lines drawn from any point in its circumference to the four angular points of the parallelogram is constant. 151. The squares on the sides of a quadrilateral are together greater than the squares on its diagonals by four times the square on the straight line joining the middle points of its diagonals. 152. In AB the diameter of a circle take two points C and D equally distant from the centre, and from any point E in the circumference draw EC, ED: shew that the squares on EC and ED are together equal to the squares on AC and AD. 153. In BC the base of a triangle take D such that the squares on AB and BD are together equal to the squares on AC and CD, then the middle point of AD will be equally distant from B and C. 154. The square on any straight line drawn from the vertex of an isosceles triangle to the base is less than the square on a side of the triangle by the rectangle contained by the segments of the base. 155. A square BDEC is described on the hypotenuse BC of a right-angled triangle ABC: shew that the squares on DA and AC are together equal to the squares on EA and AB. 156. ABC is a triangle in which C is a right angle, and DE is drawn from a point D in AC perpendicular to AB: shew that the rectangle AB, AE is equal to the rectangle AC, AD. 157. If a straight line be drawn through one of the angles of an equilateral triangle to meet the opposite side produced, so that the rectangle contained by the whole straight line thus produced and the part of it produced is equal to the square on the side of the triangle, shew that the square on the straight line so drawn will be double the square on a side of the triangle. 158. In a triangle whose vertical angle is a right angle a straight line is drawn from the vertex perpendicular to the base: shew that the square on this perpendicular is equal to the rectangle contained by the segments of the base. 159. In a triangle whose vertical angle is a right angle a straight line is drawn from the vertex perpendicular to the base: shew that the square on either of the sides adjacent to the right angle is equal to the rectangle contained by the base and the segment of it adjacent to that side. 160. In a triangle ABC the angles B and C are acute: if E and F be the points where perpendiculars from the opposite angles meet the sides AC, AB, shew that the square on BC is equal to the rectangle AB, BF, together with the rectangle AC, CE. 161. Divide a given straight line into two parts so that the rectangle contained by them may be equal to the square described on a given straight line which is less than half the straight line to be divided. III. 1 to 15. 162. Describe a circle with a given centre cutting a given circle at the extremities of a diameter. 163. Shew that the straight lines drawn at right angles to the sides of a quadrilateral inscribed in a circle from their middle points intersect at a fixed point. 164. If two circles cut each other, any two parallel straight lines drawn through the points of section to cut the circles are equal. 165. Two circles whose centres are A and B intersect at C; through C two chords DCE and FCG are drawn equally inclined to AB and terminated by the circles: shew that DE and FG are equal. 166. Through either of the points of intersection of two given circles draw the greatest possible straight line terminated both ways by the two circumferences. 167. If from any point in the diameter of a circle straight lines are drawn to the extremities of a parallel chord, the squares on these straight lines are together equal to the squares on the segments into which the diameter is divided. 168. A and B are two fixed points without a circle PQR; it is required to find a point P in the circumference, so that the sum of the squares described on AP and BP may be the least possible. 169. If in any two given circles which touch one another, there be drawn two parallel diameters, an extremity of each diameter, and the point of contact, shall lie in the same straight line. 170. A circle is described on the radius of another circle as diameter, and two chords of the larger circle are drawn, one through the centre of the less at right angles to the common diameter, and the other at right angles to the first through the point where it cuts the less circle. Shew that these two chords have the segments of the one equal to the segments of the other, each to each. 171. Through a given point within a circle draw the shortest chord. 172. O is the centre of a circle, P is any point in its circumference, PN a perpendicular on a fixed diameter: shew that the straight line which bisects the angle OPN always passes through one or the other of two fixed points. 173. Three circles touch one another externally at the points A, B, C; from A, the straight lines AB, AC are produced to cut the circle BC at D and E: shew that DE is a diameter of BC, and is parallel to the straight line joining the centres of the other circles. 174. Circles are described on the sides of a quadrilateral as diameters: shew that the common chord of any adjacent two is parallel to the common chord of the other two. 175. Describe a circle which shall touch a given circle, have its centre in a given straight line, and pass through a given point in the given straight line. 1 |