Ex. 39. ABCD is a quadrilateral; the diagonals AC and BD intersect in O, and the sides AB, CD produced meet in P; Q is a point in PB such that PQ=AB, and R is a point in PC such that PR CD. Prove that the area of the triangle PQR is equal to the difference between the areas of the triangles OAD and OBC.

Ex. 40. The line CF cuts the side AB of the triangle ABC in a point F such that AF: FB=n: 1 and lines are drawn through A and B parallel to the opposite sides. Show that the ratio of the area of the triangle formed by these lines and CF to the area of the triangle ABC is (1 - n)2 : n.

Ex. 41. Two parallelograms ABCD, A'BC'D' have a common angle at B. If AC' and DD' meet in O, prove that the ratio of OD' to OD is equal to the ratio of the areas of the parallelograms.

Ex. 42. Show how to construct a triangle having given the lengths of two of its sides and the length of the bisector (terminated by the base) of the angle between them.

Ex. 48. Prove that, if the bisector of the external angle at A, made by producing one of the sides of the triangle, meet the base in E,

BE.CE=AE2+BA. AC.

Ex. 44. Divide a line into two parts such that the square on one part is equal to the rectangle contained by the whole line and the other part.

Divide a line into two parts such that the square on one part shall be equal to three times the square on the other.

Ex. 45. Squares ADMB, BENC, CFLA are described on the sides of a triangle ABC and outside the triangle. Show that, if the sum of the squares on DL and EM is equal to five times the square on AB, then the angle ACB is a right angle.

Ex. 46. If ABFG, ACKH be the squares described on the sides AB, AC of a triangle ABC of which the angle A is a right angle, prove that BK, CF intersect on the perpendicular from A on BC.

Ex. 47. ABCD is a quadrilateral with AB=AD=4 in., CB=CD=2 in., LA=40°. E is taken on AD such that DE=1 inch and CDEF is a quadrilateral with CD=CF=2 in., DE=EF=1 in., F being within the quadrilateral ABCD. Prove (i) that the quadrilaterals ABCD and CDEF are similar, (ii) that the 4FBD = / DCE, (iii) that BF produced is perpendicular to AD.

Ex. 48. The squares ACDB, AEFG are such that E lies on AB and the square on AE is equal to the rectangle AB, EB; the squares are external to one another and FE produced meets CD in H and GD in K. Prove that FKDB and GAHK are equal parallelograms. Show also that, if AH meets CF in O, then BO, FA and CD are concurrent.

Ex. 49. From the extremities A, B of a diameter of a circle whose centre is O chords AD, BC are drawn intersecting outside the circle at P, and CA, DB, produced if necessary, intersect at Q. Prove that the directions of AB and PQ are at right angles to each other, and that the circle through C, D and O passes through the intersection of AB and PQ.

Ex. 50. The straight lines drawn through the extremities of a fixed chord of a circle to a variable point on the circle intersect the diameter of the circle which is perpendicular to the chord in P and Q. Prove that, if C is the centre of the circle, the rectangle CP. CQ is constant.

Ex. 51. From a point P, a tangent PA and a chord PBC are drawn to a circle. The bisector of the angle APB meets AB in H and AC in K; show that

AH2 HB. KC.

Ex. 52. A straight line AB is divided at C so that

AC2 AB. BC.

A circle described with centre A and radius AB cuts a circle described with centre B and radius equal to AC at D and E. Prove that CD=CE=AC, and without measurement find the sizes of the angles of the triangle ABD.

Ex. 53. Draw any straight line AB. With A as centre and AB as radius describe a circle. With B as centre and AB as radius mark off a point C on the circle, with C as centre and the same radius mark off a point D on the circle, and with D as centre and the same radius mark off E on the circle. Then prove that E is the other extremity of the diameter of the first circle.

With A and B as centres and BE as radius describe two arcs of circles to intersect at F. With F as centre and FA or FB as radius describe a circle to meet at G the circle with A as centre and AB as radius. Prove that the triangles FAG and EGA are similar, and hence that EG=AG.

From these results show how it is possible to bisect a given straight line using the compasses only.

Ex. 54. Through one of the points of intersection of two given circles any line is drawn which cuts the circles again in P, Q. Prove that the middle point of PQ is on a circle whose centre is midway between the centres of the given circles.

Ex. 55. PQ is a chord of a circle subtending a right angle at a fixed point O inside the circle whose centre is C. If ON is the perpendicular from O on PQ, and CL is the perpendicular from C on PQ, show that L and N lie on a fixed circle whose centre is the middle point of OC.

Ex. 56. One of the internal common tangents of two circles touches the circles at P and Q and meets the external common tangents in A and B. Prove AP=QB.

Ex. 57. The vertices B and C of a triangle are fixed and the angle A is given. Show that if BM and CN are perpendiculars from B and C on the opposite sides, then MN is of constant length and touches a fixed circle.

Ex. 58. Three points, P, Q, R, are taken on the sides BC, CA, AB respectively of a triangle ABC. Show that the circles which circumscribe the triangles QAR, RBP, PCQ meet at a point O. Show also that the angle BOC is equal to the sum of the angles BAC and QPR.

Draw an equilateral triangle of which each side is 3 inches and call it ABC; construct the points P, Q, R on BC, CA, AB respectively so that the angles of the triangle PQR shall be 90°, 60°, 30° (in this order) and so that PC is 1 inch.

Ex. 59. ABCD is a quadrilateral inscribed in a circle S. AC and BD meet in E, AB and DC in F. If a circle can be drawn to touch the sides of the quadrilateral FBEC, prove that its centre must lie on S.

Ex. 60. A circle S passes through the centre of another circle S'; show that their common tangents touch S in points lying on a tangent to S'.

Ex. 61. Two variable circles intersect in a given point O, and their centres lie on fixed lines through O. Prove that their common tangents always intersect on one of two fixed straight lines through O.

Ex. 62. Two circles cut at A, B; draw a circle which shall touch these two circles in such a way that the line joining the points of contact shall pass through A or B.

Ex. 63. ABC is a triangle inscribed in a circle. AP is a chord of the circle which bisects BC, and the tangent at P meets BC in N. Prove that

NC: NB=AB2; AC2.

INDEX-LIST OF DEFINITIONS

Acute angle, obtuse angle, reflex angle. An angle less than a right angle
is said to be acute; an angle greater than a right angle and less than two right
angles is said to be obtuse; an angle greater than two and less than four right
angles is said to be reflex.

Acute-angled triangle. A triangle which has all its angles acute is called
an acute-angled triangle.

Adjacent angles. When three straight lines are drawn from a point, if one
of them is regarded as lying between the other two, the angles which this line
makes with the other two are called adjacent angles. p. 4.

Adjacent sides of a polygon are sides which meet at a vertex of the polygon.
Algebra in Geometry. p. 61.

Alternate angles. p. 6.

Alternate segment. p. 97.

Altitude. See parallelogram, triangle.

Ambiguous case. pp. 24, 29 (Ex. 53), 117 (Ex. 4).

Angle. When two straight lines are drawn from a point, they are said to
form, or contain, an angle. The point is called the vertex of the angle, and the
straight lines are called the arms of the angle.

Angle in a segment. An angle in a segment of a circle is the angle subtended
by the chord of the segment at a point on the arc. p. 92.

Apollonius' circle. p. 139.

theorem. p. 67.

Arc of a circle. A portion of the (circumference of the) circle. An arc is
called major or minor according as it is greater or less than half the circum-
ference. p. 78.

Area of parallelogram. p. 53.

rectangle. p. 52.

similar triangles. p. 117.

triangle. p. 53.

Base. See parallelogram, triangle.

Bisect. Divide into two equal parts. pp. 32, 33.

Centroid. p. 136, Ex. 4.

Ceva's theorem. p. 137, Ex. 19.

Chord of a circle. The straight line joining two points on a circle.

Circle. A circle is a line, lying in a plane; such that all points in the line
are equidistant from a certain fixed point, called the centre of the circle. The
fixed distance is called the radius of the circle.

Circles touching. p. 82.

Circumcentre. The centre of a circle circumscribed about a triangle is
called the circumcentre of the triangle. pp. 84, 136 (Ex. 1).

Circumference of a circle=circle as defined above, where the term 'circle'
is used loosely as denoting the figure enclosed.

Circumscribed circle. p. 84.

Circumscribed polygon. p. 86.

Circumscribed quadrilateral. p. 89, Ex. 40.

Common tangents, exterior and interior. p. 100.

Concyclic. pp. 93, 101.

Congruent. Figures which are equal in all respects are said to be congruent..
p. 18.

Congruent triangles. p. 18.

Construct and draw, use of terms. p. 30.

Constructions. pp. 3, 30.

circle through two points to touch given line. p. 122, Ex.7.

circumcircle. p. 84.

circumscribed polygon. p. 86.

common tangent. p. 100.

constant angle locus. p. 98.

division of straight line. pp. 48, 124.

escribed circle. p. 86.

fourth proportional. p. 125.

hexagon. p. 84.

inscribed circle. p. 85.

inscribed polygon. p. 84.

mean proportional. p. 126.

pentagon. p. 142.

segment to contain given angle. p. 98.

similar figure. p. 126.

tangent. p. 99.

triangle equivalent to quadrilateral. p. 55.

triangle inscribed in circle. p. 99.

Contact of circles. p. 82.

Converse. p. 5.

Convex. p. 14.

Corollary. p. 4.