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 whoso vertical angle is a light angle a straight line is drawn from the vertex perpendicular to the base: show that the square on this perpendicular ia 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 BG 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 termmated by the circles: shew that DE and FG are equal.
