Paper 15.

1. If ABCD is a parallelogram in which the side BC is fixed and the length of the diagonal CA is given, show that the locus of D is a circle whose centre is a point O on BC produced so that BC=CO.

2. Draw a triangle ABC, right-angled at A, such that AB=4 cm. and AC=8 cm. On AB, BC, CA draw squares ABEF, BCGH, CAJK, outside the triangle ABC, the letters being taken in order round each square. On GH draw a right-angled triangle GMH outside the square BCGH equal to ABC, such that GM AB. Join FJ, EA, AK, AM. Prove that EA, AK are one straight line. Now assume that two quadrilaterals are congruent if three sides of one of them and the two angles contained by these sides are equal to the corresponding parts of the other. And prove in your figure that ABHM is congruent with EBCK, and that ACGM is congruent with KJFE. Thence prove that the square on BC is equal to the sum of the squares on BA, AC.

[You may find it convenient to outline parts of your figure with coloured pencil, and to refer to these parts by their colour.]

3. Use the first of the above figures to give a proof of Pythagoras' theorem. 4. Use the last two figures to give another proof of Pythagoras' theorem.

5. To two circles, centres O and O', an internal and an external common tangent are drawn, meeting in P. Prove that P lies on the circle on OO' as diameter.

6. ABCD is a square whose diagonals AC, BD meet at O. Circles are drawn through ABCD and AOD. A line from A meets the circles at P and Q. Prove that the triangle PQD is isosceles.

7. The tangent at T and a chord XY of a circle meet at O external to the circle: show that the internal bisectors of the angles XTY, XOT are at right angles.

8. O is the centre of a circle, OA a radius, AQ any chord. A circle through O, A, and any point C on the circumference cuts AQ in P. Prove that CP PQ.

9. A triangle ABC is right-angled at C, CD is drawn perpendicular to AB and the internal bisector of the angle C meets AB at E. Prove that

10. If through a point on the line joining the centres of two circles another straight line can be drawn such that the lengths of the chords intercepted on it by the circles are in the ratio of their respective radii, show that every other straight line through the point will possess the same property.

Two circles of radii 2 cm. and 3 cm. have their centres 6 cm. apart. A straight line is drawn through a point on the line of centres distant 3 cm. from the centre of the smaller circle and 9 cm. from the centre of the larger circle, so that the lengths of the chords intercepted on it by the two circles are equal. Find the length of these equal chords.

S. H. T. G.

11

MISCELLANEOUS EXERCISES*

Ex. 1. Prove that a quadrilateral which has two opposite sides equal and two opposite obtuse angles equal is a parallelogram.

Show that the quadrilateral is not necessarily a parallelogram if the equal angles are acute.

Ex. 2. Prove that the sum of the perpendiculars from a point inside an equilateral triangle on the sides is constant.

Ex. 3. The angles of a parallelogram are bisected externally; prove that the bisectors form a rectangle whose diagonals are equal to the sum of two sides of the parallelogram.

Ex. 4. Through a given point O draw three straight lines OA, OB, OC of given lengths so that A, B, C may be collinear and AB=BC.

Ex. 5. Two equivalent parallelograms ABCD, APQR have a common angle at A; prove that QC is parallel to DP.

Ex. 6. The points D, E, F bisect the sides BC, CA, AB respectively of the triangle ABC. FG is drawn parallel to BE to meet DE produced in G. Prove that the area of ▲ FCG is of that of ▲ ABC.

Hence prove that the area of the triangle whose sides are equal to AD, BE, CF is of that of the ▲ ABC.

Ex. 7. A point A moves along a straight line 7 and is joined to two fixed points B and C such that BC is parallel to 1; CA, BA meet a given straight line parallel to l in points E, F. Show that the intersection of BE, CF describes a straight line.

Ex. 8. ABC is any triangle. Show how to inscribe a square PQRS in the triangle so that P lies on AB, Q on AC, and the side RS on BC.

Ex. 9. ABC is a triangle right-angled at A; BDEC is a square on the remote side of BC; if AD, AE cut BC in P, Q respectively, show that PQ is a mean proportional between BP and QC.

Ex. 10. Prove that the bisectors of the angles of any quadrilateral intersect in four concyclic points.

Ex. 11. Prove that the bisectors of opposite angles of a cyclic quadrilateral meet the circle again in points which are the opposite extremities of the same diameter.

* These Exx. are arranged in four groups; the first and third groups do not involve circles; the Exx. in the last two groups are harder than those in the first two groups.

Ex. 12. Show that the centres of the squares described on the hypotenuse of a right-angled triangle are each equally distant from the two sides containing the right angle, and determine the lengths of these distances in terms of the sides.

Ex. 13. ABCD is a quadrilateral inscribed in a circle and AD, BC produced intersect in P. Prove that PC. AB=PA.CD.

Mention any two ratios each of which is equal to the ratio of the area of As PAB, PCD.

Ex. 14. AB is a diameter and PQ a parallel chord of a circle; O is any point in AB. Prove that OP2+OQ2=OA2+OB2.

Ex. 15. A triangle ABC is inscribed in a circle and chords AX, BY are drawn parallel to the sides BC, CA. Prove that XY is parallel to the tangent at C.

Ex. 16. ABC is a triangle inscribed in a circle. The tangents at B, C meet in P; and PQ, parallel to AB, meets AC in Q. Prove that BPQC is a cyclic quadrilateral.

Ex. 17. Two circles cut in A and B; P is a point on one of the circles; the tangent at A to that circle cuts the other circle in Q, and BP cuts it in R. Prove that QR is parallel to AP.

Ex. 18. H is the orthocentre of a triangle ABC; HA is produced to K so that HK=BC; KM is drawn parallel to HC to cut BH produced at M. Prove that AMKHA ABC.

Ex. 19. The perpendiculars from A, B, C on the opposite sides of triangle ABC meet the circumcircle in A', B', C' respectively. Prove that the lines drawn from O, the centre of the circumcircle, to A, B, C are perpendicular respectively to the sides of the triangle A'B'C'.

Ex. 20. A variable circle passes through a fixed point A and cuts at right angles a given circle whose centre is O. Prove that the locus of the centre of the first circle is a straight line perpendicular to OA.

Ex. 21. Two circles touch externally at A. Any line through A meets the circles in P, Q. PT, a tangent to the circle AQ, meets the circle AP again in R. Prove that AT bisects the angle RAQ.

Ex. 22. The opposite sides of a quadrilateral inscribed in a circle meet in P and Q. Prove that the bisectors of the angles at P and Q form a rectangle.

Ex. 23. Prove that, if chords AA', BB', CC' of a circle are concurrent, the products BC'. CA'. AB' and CB'. AC'. BA' are equal.

Points of the compass are marked round the circumference of a circle and lines are drawn from the points N., N.N.E., N.E. to the points E.S.E., S., W. respectively. Show that these lines are concurrent.

Ex. 24. AB is a chord of a circle, C its middle point; D and E are points on the circle such that ▲ ACD= LACE. Prove that ▲ DAG= LAEC.

Ex. 25. A, P, Q are three points on a circle such that the angle PAQ is given; find the envelope of PQ (i) when A is fixed, (ii) when A takes all positions on the circle.

Ex. 26. Through a fixed point A on a given circle a line is drawn cutting the circle in P, and AP is produced to Q so that PQ is of constant length. Show that the line through Q perpendicular to AQ touches a circle.

Ex. 27. Two pairs of intersecting straight lines meet to form four triangles. Show that the circumcircles of these four triangles all meet in one point.

Ex. 28. The bisector of the vertical angle A of a triangle ABC meets the base BC in E. Show that the ratio of the diameters of the circumcircles of the triangles ABE, ACE is AB; AC.

Ex. 29. ABCD is a quadrilateral inscribed in a circle, to which the tangents at A, B, C and D form a quadrilateral PQRS. If PQRS is cyclic, prove that AC is perpendicular to BD.

Ex. 30. In an isosceles triangle ABC, AB is one inch long, and the angles at A and B are each double the angle at C. A circle is described round ABC, and a tangent to this circle at A meets CB produced in D. Find by calculation, or by construction, the lengths AD, BD and the angle at D. (See pp. 142–145.)

Ex. 31. The sum of the rectangles contained by opposite sides of a cyclic quadrilateral is equal to the rectangle contained by its diagonals. (See p. 128, Exx. 33, 41.)

(Ptolemy's theorem. This Ptolemy is the astronomer who lived in Roman Alexandria in the second century A.D., 400 years after Euclid. He made great advances in trigonometry.)

Ex. 32. P is any point on the minor arc BC of the circumcircle of an equilateral triangle ABC. Prove that PB+PCPA.

Ex. 38. Construct the quadrilateral, of which are given the lengths of the diagonals and of the straight line joining the middle points of one pair of opposite sides.

Ex. 34. ABCD is any quadrilateral. Prove that the locus of a point P which moves so that the sum of the areas of As ABP, CDP is constant, is a straight line.

Ex. 35. Three parallel lines are cut by two transversals in points A, B, C ; A', B', C'. Show that AB. CC' + BC. AA' + CA. BB'=0.

Ex. 36. ABCD is a quadrilateral in which AB-CD. Prove that the line joining the middle points of the sides BC and AD is equally inclined to the other two sides.

Ex. 37. If from a point P perpendiculars PL, PM, PN are drawn to the sides BC, CA, AB of a triangle ABC respectively, show

(i) that BL2+CM2+AN2=CL2+AM2+BN2,

(ii) that the perpendiculars drawn from A, B, C to MN, NL, LM respectively meet at a point.

Ex. 38. If the opposite sides AB, CD of a quadrilateral meet in P, and if G, H are the mid-points of AC, BD, prove that the area of the triangle PGH is one-quarter of the area of the quadrilateral ABCD.