4. PQR is a triangle right-angled at Q, S is the mid-point of PQ; prove that PR2 RS2+3QS2.

5. A circle touches a second circle externally, and the tangents to it from the centre of the second circle contain an angle of 60°. Show that the circles have equal radii.

6. Two circles intersect at P. AB is a diameter of one circle and PA, PB (produced if necessary) meet the other circle in C and D. Show that CD is a diameter of this circle.

7. If one circle touch another internally at O, and any straight line cut the outer circle at A, D and the inner at B, C, prove that AOB = < COD.

8. Prove that the area of the regular hexagon inscribed in a circle is twice the area of the inscribed equilateral triangle. Verify this fact by cutting a regular hexagon out of paper, and folding it.

9. At two points A, B of a straight line perpendiculars AC, BD are erected and AD, BC meet in a point E; from E a perpendicular EF is drawn to AB. Prove that

10. ABCD is a rectangle inscribed in a given circle. From a point P in the circumference of the circle, perpendiculars PH, PK, PM, PN are drawn to the four sides AB, BC, CD, DA. Show that PH.PM PK.PN.

Paper 4.

1. A triangle ABC is such that the angle at A is equal to the sum of the angles at B and C. Prove that a point D can be found in BC such that the triangles DAB, DAC are both isosceles.

2. Perpendiculars are drawn from the corners of a rectangle and from the intersection of the diagonals to any straight line outside the rectangle. Prove that the sum of the lengths of the perpendiculars from the vertices is four times the length of the perpendicular from the intersection of the diagonals. Is the same property true of a parallelogram?

3. Call the corners of a rectangular sheet of paper A, B, C, D (AB being a long side of the rectangle); if it were folded along the diagonal AC, then AB and CD would cut at a point we will call O. Make a freehand sketch of the figure you would obtain and prove triangles BCO, ADO (in the folded figure) equal in area.

4. If from a point O within the triangle ABC perpendiculars OX, OY, OZ are drawn to the sides BC, CA, AB respectively, show that the sum of the squares on BX, CY, and AZ is equal to the sum of the squares on CX, AY, and

BZ.

5. AB is an arc of a circle and C its middle point. Prove that the angle ABC is one-quarter of the angle which the arc AB subtends at the centre of the circle.

6. Four points A, B, C, D are taken on a circle so that the chords AC, BD are at right angles to each other. Prove that the tangents at A, B, C, D form a quadrilateral which can be inscribed in a circle.

7. In the figure, BC touches the circle ABD. Show that CE touches the circle ADE at E. (You may assume the converse of the "alternate segment" theorem.)

8. C is a fixed point on a fixed line OA; P C is a point which moves on a fixed circle whose centre is A, and a point Q is taken on OP such

9. AB the diameter of a circle is 2". Draw AE 4" long cutting the circle at C. Produce AB and draw ED perpendicular to AB. Prove that the area CBDE is three times the area of the triangle ABC.

10. Three points A, B, C are taken upon a straight line, AB being greater than BC; similar and similarly situated quadrilaterals ABEF, BCGH (lying upon the same side of the line ABC) are described upon AB and BC. Prove that EG and FH meet AC produced in the same point X.

Prove that the line joining the intersections of the diagonals also passes through X.

Paper 5.

1. In OX, OY, show how to find points A, B such that LOAB=3LOBA. Give a proof. [What angle is equal to the sum of these angles?]

2. Show how to draw a straight line equal and parallel to a given straight line and having its ends on two given straight lines.

3. A straight line is drawn parallel to the base BC of a triangle ABC cutting AB at X and AC at Y; prove (i) that triangles XBC, YBC are equal in area, and (ii) that triangles ABY, ACX are equal in area.

4. In a triangle ABC, the angle C is 90°. AC is produced to D, so that CD=CB. Show that the square on AD= the square on AB+4 times the area of the triangle.

5. If ABCD is a square circumscribing a circle, and any tangent to the circle meets AB, BC, CD, AD in P, Q, R, and S respectively, prove that PQ, QR, RS subtend equal or supplementary angles at the centre of the circle.

6. PQRS is a quadrilateral inscribed in a circle. X, Y are points on PS, QR respectively such that XY is parallel to SR. Prove that a circle can be drawn to pass through P, Q, Y, X.

7. ABC is an acute-angled triangle with AB equal to AC. In AB a point D is taken so that CD=CB. Prove that the circle circumscribing the triangle ADC touches BC.

[See next page

8. Two circles touch internally at X. From A a point on the outer circle tangents AP, AQ are drawn to the inner: show that the angle PAQ is least when A is diametrically opposite to X.

9. ABCD is a rectangle. A straight line meets AB and CD in P and Q, and a perpendicular straight line meets AD and BC in R and S. Show that

PQ: RS=AD: AB.

10. CD is a chord at right angles to the diameter AB of the circle ABCD; E is any point in CD, and AE is produced to meet the circle in F; prove that CE: ED=CF: FD.

Paper 6.

1. ABCD is a parallelogram; E is the mid-point of AB; CE and DA are produced to meet at F. Prove AF=AD.

2. ABC is a triangle obtuse-angled at C. AC, BC are bisected at D and E. Through D, E perpendiculars DF, EG are drawn to AC, BC meeting AB in F, G. Show that the perimeter of the triangle CFG is equal to the side AB, and that the angles GFC, CGF are double of the angles at A and B respectively. Deduce a construction for a triangle when the perimeter and two angles are given.

3. ABCD is a parallelogram, P a point in AB and Q a point in DC produced so that AP=CQ. Prove that the figure PBQD is double the triangle ADB.

4. The perpendicular AX on the base BC of a triangle cuts the base internally at X, so that BX=3XC; prove that 2AB2=2AC2 + BC2.

5. Prove that if AB, BC, CD are three equal chords in a circle, then the quadrilateral ABCD is a trapezium.

6. Two circles of radii a and b intersect one another and 2c is the length of their common chord. If the line of centres cuts the circles at X and Y, calculate XY.

7. T is any point outside a circle ABC whose centre is the point O. Through T two lines are drawn, TA touching the circle and TBC cutting it. If M is the mid-point of BC, show that LAMT LAOT.

8. Two circles intersect at the points A, B, and CD, a chord of one, touches the other at the point E. Prove that the angles CAE and EBD are equal or supplementary.

9. ABC is a triangle such that the areas of similar polygons described on the sides BC, CA, AB, as corresponding sides, are respectively 17 sq. cm., 12 2 sq. cm., and 5 sq. cm. Find the size of the angle BAC, showing clearly how you arrive at your result.

10. A straight line through the vertex A of a triangle ABC meets the side BC in P and meets the circumcircle of the triangle again in Q; a parallel through Q to BC meets AB, AC in R, S respectively. Prove that

AQ2: RQ. QS=AP: PQ.

Paper 7.

1. From a point C within the acute angle formed by two lines OA, OB a line is drawn parallel to AO to meet OB in B. The circle whose centre is C and radius CB cuts OB again in D; DC produced meets OA in A. Prove that OA is equal to AD.

2. Draw a triangle ABC having AB=6 cm., BC=7 cm., ▲ B=43°. If D is a point in BC such that ADC=70°, prove (without any measurement) that AD must be less than AC. Verify by measurement.

3. Through the vertices of a triangle ABC, parallel straight lines are drawn to meet the opposite sides of the triangle in points X, Y, Z; prove that

Δ ΧΥΖ=2Δ ABC.

4. In the right-angled triangle ABC, BC is the hypotenuse and the side AB is double the side AC. A square is described on BC and is divided into two rectangles by a line through A perpendicular to BC. Prove that one rectangle is four times the other.

5. The tangents at B and C to the circumcircle of a triangle ABC meet in O; and the circle whose centre is O and radius OB or OC meets AC produced in Q. QO is drawn to meet AB in R; prove that R lies on the circle BCQ.

6. Through two given points P, Q on a circle draw a pair of equal and parallel chords.

7. Show how to draw two concentric circles, one of which passes through two given points A, B, while the other touches a given circle at a given point P, and explain in what case the construction is impossible.

8. Two circles touch externally at O; AB, CD are their common tangents. Prove that ABDC is a trapezium. Show also that the length of the common tangent at O intercepted between AB and CD is half the sum of AC and BD.

9. PQR is an isosceles triangle whose vertex is R, and the base PQ is produced to any point O. Prove that if OR cut the circle circumscribing the triangle in S,

OS.OR OR2 – QR2.

10. PP', QQ' two chords of a circle intersect at an external point O. If

show that the chords are equidistant from the centre.

Paper 8.

1. PQR is an isosceles triangle having PQ=PR. A straight line is drawn perpendicular to QR and cuts PQ, PR (one of them produced) in X, Y. Prove that the triangle PXY is isosceles.

2. Show how to construct a rhombus PQRS having its diagonal PR in a given straight line and its sides PQ, QR, RS passing through three given points L, M, N respectively. Give a proof.

3. The sides BC, CA, AB of a triangle are produced to D, E, F, so that CD=BC, AE=CA, BF=AB. Prove that the area of DEF is seven times the area of ABC.

y

4. Ox, Oy are perpendicular straight lines, ABCD is a square. If OA=a, OB=b, find the distances from Ox and Oy of the points C, D and the centre of the square, i.e. the point of intersection of AC and BD. Hence, or otherwise, prove By that, if the square is moved so that its vertices A and B always remain on Ox and Oy, the centre of the square b describes a straight line.

5. A circle whose centre is O is touched internally at A by a circle of half its radius. A radius OQ of the

G

E

former circle cuts the smaller circle at P. Prove that arc AQ=arc AP.

FX

6. OADB is a rectangle; ACB is a triangle, right-angled at C, described on a diagonal AB of the rectangle. Prove that the angle between CD and OA is equal to the angle CAB.

7. ABCD is a quadrilateral in a circle. One side BC is produced to E. Prove that the bisectors of the angles BAD, DCE meet on the circumference.

8. ABCD is a cyclic quadrilateral and the tangents at B and D meet in O. Prove that BOD is equal to the difference between BCD and ▲ BAD.

9. Find the locus of a point which moves so that its distance from a given line is half its distance from a given point on the line.

10. Prove that, if circles are described passing through two given points A and B and cutting a given circle in P and Q, the chord PQ cuts AB in a fixed point.

Paper 9.

1. Show how to draw a straight line equidistant from A and B and making a given angle with AB.

2. OPQRS is a pentagon in which OP=OS, PQRS, and the diagonals OQ, OR are equal; prove that the diagonal PS is parallel to QR.

3. ABCD is a parallelogram; BC is bisected at E, and CD at F; AE and DC are produced to meet at G, and AF and BC are produced to meet at H. Prove that the area of the triangle AGH is equal to one and a half times the area of the parallelogram.