II. HILARY TERM, 1905. 1. (b) The side BC of a triangle ABC is produced to D, and the angles ACB, ACD are bisected by the straight lines CE, CF respectively. Prove that EČF is a right angle. 2. (b) Two isosceles triangles have respectively (1) each of the base-angles double of the vertical angle, and (2) the vertical angle double of each base-angle. Prove that the vertical angles are to one another in the ratio 2 : 5. 3. (6) Prove that the straight line joining the middle points of two sides of a triangle is parallel to the third side. 4. Describe a rectangle having an area (1) double of, (2) equal to the area of a given triangle. 7.7) Prove that the straight line which passes through the middle points of two parallel chords of a circle is a diameter of the circle. 8. (6) If two circles touch, prove that the distance between the centres is equal to either the sum or the difference of the radii. 9. (6) A variable straight line passes through a fixed point. Find the locus of the foot of the perpendicular drawn to it from another fixed point. 10. Two circles intersect in the points A, B. Show how to draw a straight line to touch both circles, and prove that the middle point of the distance between the points of contact lies on the straight line AB. III. TRINITY TERM, 1905. 1. (6) Show that the sum of the sides of a triangle is greater than the sum of the lines joining its angular points to the middle points of the opposite sides. 2. State and prove a construction for the trisection of a right angle. 3. Construct a triangle having given the base, one of the angles at the base, and the sum of the remaining sides. these the segments in a circle, if to 5. (6) Show that if the opposite sides of a quadrilateral are equal, the figure is a parallelogram. 6. In the triangle ABC, AB is equal to AC and BD is drawn perpendicular to AC cutting AC in D. Show that the square on BC is equal to twice the rectangle AC.CD. 8. (6) Prove that in a circle, if two equal chords intersect, the segments of the one are equal respectively to the segments of the other. 9. (b) Show that if two circles intersect, the tangents drawn to them from any point on the common chord are equal. 10. Describe a circle which shall pass through two fixed points and have its centre on a given straight line. What are the cases in which (1) no such circle can be drawn; (2) an indefinite number of such circles can be drawn? IV. SEPTEMBER, 1905. 2. A is a fixed point in a straight line AB; P is a point outside the line AB; show how to draw a line PC cutting AB in C, such that PC = AC, 4. Prove that: (1) the line drawn through the middle point of a side of a triangle parallel to another side bisects the third side ; (2) every right-angled triangle is divided into two isosceles triangles by a straight line drawn from the right angle to the middle point of the base. 5. (6) Construct a rectilineal figure equal to a given rectilineal figure and having fewer sides by one than the given figure. 6. (6) Show that the sum of the squares on the sides of a parallelogram is equal to the sum of the squares on the diagonals. 7. Prove that the line joining the middle points of two parallel chords of a circle passes through the centre. 8. Find the locus of the centres of the circles which touch two given straight lines. 9. (6) Two circles intersect at P and Q; through X, a point on the circumference of one of them, straight lines are drawn passing through P and Q, and cutting the other circle again in R and S; show that RS is parallel to the tangent at X. 10. O and P are the centres of two circles intersecting at A and B; Q is the middle point of OP; through Ď a line is drawn perpendicular to QB, cutting the circles again in X and Y. * Show that XB = YB. and that p V. MICHAELMAS TERM, 1905.. 1. (b) On the sides AB and BC of a triangle ABC, equilateral triangles ABP and BCQ are described, so that P and C are on opposite sides of AB, and Q and A on opposite sides of BC. Prove that AQ and CP are equal. 2. If ACB, ADB are two triangles on the same side of AB, such that AC is equal to BD and AD is equal to BC, and if AD and BC intersect in 0, show that the triangle AOB is isosceles. 3. (b) Describe a parallelogram equal in area to a given square, such that the sum of the sides of the parallelogram shall be double the length of the sum of the sides of the square. 4. (b) The triangle BAC has the angle A a right angle; from Ď the middle point of AB a perpendicular DX is drawn to BC. Prove that the difference of the squares on CX and BX is equal to the square on CA. 6. With a given point as centre describe a circle, such that the points in which it meets a given circle shall be the ends of a diameter of the given circle. 7. (6) Show how to construct the common tangents of two circles which intersect one another. (c) Show also that the line joining the points of intersection of the two circles will bisect the two common tangents. 8. (6) From two points P, Q, on the circumference of a circle, are drawn two parallel lines PA and QB to meet the circumference in A and B respectively, and also two other parallel lines PX and QY to meet the circumference in X and Y respectively. Show that AY is parallel to BX. VI. HILARY TERM, 1906. 1. (6) In a triangle ABC, the angles B and C of the triangle are bisected by straight lines which intersect one another at 0. Prove that, if BO is equal to CO the triangle ABC is isosceles. 2. On a given straight line as base, construct a triangle, such that one of the base angles shall be twice, and the other three times the vertical angle. 3. Find the locus of the vertex of a triangle, when its area and base are given. 4. ABCD is a parallelogram. F, any point in the side CB, is joined to A and B, and the line DF is produced to meet AB produced in G. If CG is joined, prove that the triangles ABF and CFG are equal in area. 6. (6) Divide a given straight line into two parts so that the rectangle contained by those parts may be equal to the square on a given straight line, the length of the second given line being less than half that of the first. 8. The ends A, B of a straight line AB of given length move on two fixed straight lines 0 A, OB which are at right angles to one another. Find the locus of the middle point of the line AB. 9. Given two circles which touch internally, if another circle be drawn touching both, the sum of the distances of its centre from the centres of the two given circles is always the same. 10. (a) If from a given point 0 outside a circle a straight line O AB be drawn to cut the circle in A and B, show that the rectangle contained by 0 A and OB is the same for all lines through 0. (6) Given a point 0 outside a circle, find a point P on the circumference such that the middle point of OP shall also lie on the circumference; and show that for this to be possible, 0 must lie within another circle with the same centre as the original one, and a radius three times as great. VII. TRINITY TERM, 1906. 2. If from one extremity of the base of a given isosceles triangle a straight line is drawn perpendicular to the opposite side, show that the angle it makes with the base is half the vertical angle. 3. ABCD is a square and E, F, G, H are the middle points of AB, BC, CD, DA respectively. Show that the quadrilateral EFGH is a square and that its area is half that of ABCD. 4. Having given a triangle A BC, describe an isosceles triangle on AB as base, which shall be equal in area to ABC. 6. (6) If the straight line AC has been divided at B so that the square on AB is equal to the rectangle AC. BC, show that if AB be bisected at 0, and the square on AB be ABXY, the straight line 0X is equal to the straight line OC. 8. ABC is an equilateral triangle and on AB as diameter a circle is described. Prove that it will pass through the middle points of AC and BC. 10. Show that all points, the tangents from which to a given circle are of given constant length, lie on a circle concentric with the given circle. VIII. SEPTEMBER, 1906. 2. Construct an isosceles triangle whose vertical angle shall be four times each of the base angles. 3. On the equal sides AB, AC of an isosceles triangle ABC, equilateral triangles ABP and ACQ are described, so that P and C, and Q and B are on opposite sides of AB and AC respectively. If QC and PB meet in 0, show that the triangle OBC is isosceles. 5. ABCD is a parallelogram and 0 the point of inter |