50. AB is a fixed finite line; AX, BY are perpendiculars to it; and P is a point in it: if X, Y and P vary subject to the condition that AX. BY = AP. BP, find the Locus of the foot of the perpendicular from P on XY.

51. Given the mid points of the sides of a polygon, construct it.

2

NOTE-Let A1, A2, &c., An be mid pts. Take any pt. P1, and draw the trial pol. P, P, &c. Pn, so that A1, A2, &c., are respective mid pts. of P1 P2, P2 P1, &c. Join PnP, and bisect it in X. Draw X1 X2, X, X3, &c., so that A1, А1⁄2, &c. are their respective mid pts. Join XX1: it will go thro. Аn; and XX, &c. Xn is reqd. pol.

=

52. Four rods AB, BC, CD, DA of given commensurable lengths, such that AB + CD BC + DA, are pivoted together at A, B, C, D, so as to be capable of free angular motion in one plane; if AB is fixed, find the Locus of the in-centre of the quadrilateral formed by the rods.

53. Given two circles of a co-axal system, describe a circle of the same system to

1o, go through a given point; or

2o, touch a given line; or

3o, touch a given circle; or

4o, cut a given circle orthogonally; or

5o, cut the join of two given points harmonically.

54. Given six concyclic points A, B, C, D, E, F, find a seventh P, concyclic with them, so that the cross-ratios (PABC) and (PDEF) may be equal.

NOTE-See General Addenda ii. (6). There are two solutions, viz. the pts. in which XY meets the O.

55. Solve the last Problem when collinear is substituted for concyclic.

56. Inscribe a triangle in a given circle, so that its sides may pass respectively through three given points. (Castillon's Problem)

NOTE-If A, B, C are the given pts.; B'C', C'A', A'B' their polars; L, M, N the pts. in which A'A, B'B, C'C cut B'C', C'A', A'B'; then the sides of ▲ LMN will cut the in six pts.; and if these are joined alternately, two As solving the Prob. are obtained. For proof use General Addenda vii. (5), viii. (4), and Exercise 7, p. 376.

S

57. Circumscribe a triangle about a given circle, so that its corners may be respectively on three given lines.

58. If a triangle has one angle fixed in magnitude and position, and its perimeter is given; find the Envelope of its circum-circle. (Manheim) NOTE-Invert with respect to the Vertex of the fixed λ.

MISCELLANEOUS EXERCISES.

NOTE-There is no particular arrangement, either of difficulty or otherwise, in the following.

1. If the bisectors (terminated by the opposite sides) of two angles of a triangle are equal, prove that the sides opposite these angles are equal.

NOTE—Iƒ BX, CY are the equal bisectors, complete □XCYZ, and join BZ ; assume that ABC > ACB; thence XC (or YZ) > YB ; and .. YBZ > YB; >YZB; but XBZ XZB; .:. XBY < XZY (or YĈX) contrary to the assumption.

=

2. In Castillon's Problem (p. 382, Ex. 56) find what conditions in the data make the solution indeterminate.

3. Given in a triangle (with the notation of p. 211) the angle A, and s find the Envelope of the circum-circle.

(Manheim)

a,

4. Given the base of a triangle and the radius of its circum-circle, find the Locus of its in-centre.

5. Prove that the sum of the squares on the twelve lines from the corners of a triangle to the points of contact of its circles of contact with the corresponding opposite sides, is equal to five times the sum of the squares on the sides of the triangle.

6. Prove that the sum of the squares on the tangents from the centres of the four circles of contact of a triangle to any circle through the circumcentre, is equal to three times the square on the diameter of the circum-circle.

7. If a, b, c, d are the successive sides of an ordinary quadrilateral; show that, if a circle can be inscribed in it, the process, given in Note (1) p. 322, to form it into a cross-quadrilateral, fails; and that, in any case, if the process gives a, c, as diagonals, it will not give b, d, as diagonals; and vice versa.

8. Show that a common tangent to two circles subtends a right angle at either limiting point.

9. If A, B are inverse points, show that—

1o, for every point P on the circle of inversion, PA : PB is constant:

2o, if A is inside the circle, the segments of any chord through A subtend equal angles at B;

3o, if B is outside the circle, the segments of any chord through B subtend supplementary angles at A.

10. Show that any two circles and their inverses are touched by four circles, each of which cuts the circle of inversion orthogonally.

11. If ABC is a triangle, and points P, Q are taken in AB, AC, respectively, such that BP. BA + CQ. CA CB2; show that the Locus of the intersec

tion of BQ, CP, is a circle.

=

12. If TP, TQ are tangents to a circle; Pp, Qq a pair of parallel chords; and Tt parallel to them, cutting PQ in t; prove that

13. If ABCD is any parallelogram, and P a point within it at which opposite sides subtend supplementary angles; show that the circles PAB, PBC, PCD, PDA are all equal.

14. If on sides AB, AC, of a triangle, isosceles right-angled triangles AEC, AFB are described, both either externally or internally; and if D is the mid point of BC; prove that DEF is an isosceles right-angled triangle.

15. In the sides of a triangle ABC, respectively opposite A, B, C, points D, E, F are taken, so that

show that AD, BE, CF cut each other in the same proportion.

16. If ABCD is a cyclic quadrilateral, and E, F are points in CB, CD, such that the angles DAE, BAF are right; prove that EF goes through the centre of the circle round the quadrilateral.

17. If A, A, B, B' are four collinear points, in order; and P any point at which AA', BB' subtend equal angles; prove that

18. If APQ is the tangent at a fixed point A, on a fixed circle, and AP. AQ is constant; prove that the Locus of the intersections of the second tangents from P, Q, is a straight line parallel to APQ.

19. If ABCD is any quadrilateral, and P a point within it at which opposite sides subtend supplementary angles; then, if EF is the third diagonal, prove that the angles APC, BPD, EPF have common bisectors.

20. If A, B are fixed points, in a fixed tangent, to a fixed circle; and X, Y any harmonic conjugates to A, B; find the Locus of the intersection of tangents from X, Y to the circle.

21. Given a triangle, an area, and a ratio; draw a transversal to cut off a triangle equal to the area, and have its segments in the given ratio.

22. S, S' are inverse points with respect to a circle; CA the radius through S; H the mid point of SS'; if P is any point on the pedal of the circle with regard to S, prove that

=

NOTE—The pedal of a ©, with respect to a pt., is the Locus of the foot of the from the pt. on any tang. The form of the result shows that this pedal is what is called a focal curve, of the kind whose type is r± μr' a; where r, r' are the radii vectores from the foci S, H; and μ, a are const.: such curves are known as 'Cartesian Ovals. Taking the case when S is within the O; if PN is to SH, the result comes by eliminating SH, SN between

r'2 r2 + SH2

=

2 SH. SN, 2 b. SH

=

r).

a2 - b2, and b. SN = r (a 23. If X, Y, Z are points in sides of triangle ABC, respectively opposite A, B, C such that

24. If In, Cn are the areas of the regular in- and circum-polygons of n sides with respect to the same circle, prove that

25. Show that of two regular isoperimetrical polygons, the maximum is that which has the greater number of sides.

NOTE-Hence may be deduced that the is the maximum area of given perimeter.

26. Two sides of a given triangle touch two fixed circles, find the Envelope of the third side.

27. If two sides of a given polygon touch two fixed circles, show that all the sides touch fixed circles.

28. (1) Through fixed points A, B, a variable circle is drawn, cutting a fixed circle in X, Y; and XY, AB meet in T; if a variable line Tg G is drawn, meeting the fixed circle in g, G, prove that

AG. GB: Ag.gB

:

(2) Two points A, B, and a circle are given: show that a point T, and lines TV, TZ, can be found, such that, if from T a variable line is drawn to cut the circle in G, g, then

AG. GB: Ag.gB = TG: Tg.

CC

(Ivory)

29. A, B are inverse points with respect to a circle whose centre is C; P is any point on the circle; BN is perpendicular to the tangent at P; DCE is the diameter perpendicular to CA; PM is the perpendicular on CA (or AC produced) and meets AE in Q: prove that MQ, BN are equal.

30. Show that a variable circle, cutting two given circles at given angles, constantly touches two fixed circles, and cuts a third orthogonally.

31. If a circle constantly touches two fixed circles, show that it cuts any circle co-axal with them at a constant angle.

32. If a circle A cuts a circle B in X, Z; and touches a circle concentric with B in Y; then the arcs XY, YZ are obviously equal: derive a Theorem from this by inverting with respect to a point on the circumference of A.

33. Through each corner of a triangle ABC, parallels are drawn to the opposite sides, forming a new triangle whose sides are YAZ, ZBX, XCY; show that the nine-point circle of ABC touches the nine-point circles of XBC, YCA, ZAB at the mid points of BC, CA, AB, respectively.

34. Through a fixed point, within a fixed angle BAC, draw XPY, so that the perimeter of the triangle AXY is minimum.

35. Find the Locus of the centre of a circle cutting two given lines at given angles.

27

36. A, B, C, D, are concyclic points, in the order named: O1, O2, 03, 04, are the respective orthocentres of the triangles BCD, ACD, ABD, ABC : prove that

1o, O1 A, O2 B, O, C, O, D, are concurrent; and that,

2o, if the quadrilateral ABCD is turned, in its own plane, round this point of concurrency, through 180°, it will coincide with the quadrilateral O, O2O2 O1. 37. If C is the centre of a fixed circle, ACB an angle of fixed magnitude, and APB a tangent to the circle; show that the area ABC will be minimum when P is the mid point of the intercepted arc. Hence solve this Problem

To circumscribe about a given circle a quadrilateral of which the opposite angles shall be of given magnitude, and the area minimum.

38. Given a circle and a fixed point A, prove that another fixed point B can be found such that, if a tangent is drawn at a variable point P, AP2 will vary as the perpendicular from B on this tangent.

NOTE-Recollect that a tang. is the polar of its own pt. of cont., and use Salmon's Theorem.

39. If on the sides of any triangle equilateral triangles are described (all externally, or all internally to the triangle) show that the joins of their centres form an equilateral triangle. (Cf. pp. 318, 319.)