13. Two fixed lines meet on the circumference of a fixed circle: a variable point P is taken in one line; and its polar, with respect to the circle, meets the other in Q: show that PQ goes through a fixed point.

14. If two of the vertices of the self-conjugate triangle XYZ [Theorem (6)] lie on circles concentric with the circle round the quadrilateral; then will the third vertex lie on a concentric circle.

15. If XX', YY, ZZ' are the diagonals of a quadrilateral circumscribing a circle, centre O; and L, M, N their mid points; then the ratios of OX. OX', OY. OY', OZ. OZ', to each other, are respectively the same as those of OL, OM, ON. (W. S. McCay, Educational Times. Reprint, Vol. XXXIX. p. 88.)

16. Show that, assuming Pascal's Theorem [Section ii. Theorem (6) Note], the following Theorem (Brianchon's) can be deduced by Polars—The joins of the opposite corners of a hexagon which circumscribes a circle, are concurrent.

Def. The circle with respect to which a triangle is self-conjugate is called the polar circle of the triangle.

17. Using the extended definition of a complete quadrilateral given in the Note on p. 299; if XBA, XCD, YCB, YDA are the four lines; L, M, N the mid points of diagonals AC, BD, XY; P, Q, R the points where the diagonals intersect; O1, O2, O3, O, the ortho-centres of the four triangles XBC, XAD, YAB, YCD; and the centre of the circum-circle of PQR; then—

1o, the five points 01, 02, 03, 04, are collinear;

2o, this line of collinearity is perpendicular to LMN, and is the radical axis of the three circles whose diameters are the diagonal;

3o, the polar circles of the above four triangles, and the circle , have LMN as their radical axis;

4o, the set of three circles and the set of five circles cut orthogonally.

Λ

NOTE-Unless XAY is obtuse, 2 of the ▲s will not be obtuse-angled, and .. will not have polar circles. The proofs of the theorems depend on properties given on pp. 299, 352 (Ex. 7), 368, 370, 373.

18. Show that the harmonic section of a line by a circle, pole, and polar, may be proved by Inversion.

NOTE-In figs. of Theor. (4) take X as centre of inversion; and for the radius of inversion take in fig. (1) the tangent from X, and in fig. (2) the semichd. thro. XL to AB. Then the inverts into itself; O, Y are inverse pts.,

and C, Z are inverse pts. It will readily follow that

MISCELLANEOUS PROBLEMS.

1. Given the rectangle under two lines, and the difference of their squares, to find the lengths of the lines. (The omitted case of 11, p. 227.)

NOTE-Let AB2 be the given diff. of sqs., and ABC the given rect. Produce AB to X, so that AX. BX BC'. On AX place a semi-©; and produce CB

=

to meet it in Y. AY, BY are the reqd. lengths.

2. Bisect a given triangle by a line through a given point either within or without it.

NOTE-Let P be the pt., ABC the A. and AQ. AP = ✯ AB. AC: let segt. on

XP is the bisector.

=

CAP,

Draw AQ so that BAQ
PQ, contg. BẤP, cut AB in X;

3. Divide a triangle into parts in a given ratio, by a line parallel to a given direction.

NOTE-Let AE, || to given direc., meet BC in E: divide BC in D, in given ratio, BD being lesser part: take BX mean propl. to BE, BD: then || thro. X to AE divides ▲ as reqd.

4. From a corner of a triangle draw a line to meet the opposite side so as to be a mean proportional between the segments into which it divides that side.

5. Given three collinear points; find a point collinear with them, so that its distance from one may be a mean proportional between its distances from the other two.

6. Given three points, not collinear; find a point whose distances from the three points are proportional to given lengths.

7. Construct a triangle which shall have a given ratio to a given triangle. 8. Construct a triangle of given species, so that the distances of its corners from a given point may be equal to given lengths.

ANALYSIS-Suppose ABC the reqd. ▲ and P the given pt.: then, if on AP a A PAX is described simr. to ▲ CAB, and BX is joined, ▲s BAX, CAP will be simr., .. CA : BA CP : BX, ... BX is known, and ▲ PBX can be

constructed.

=

Exercise 93, p. 85, gives the construction for the particular case of an equi

lateral Δ.

9. Find a point within a triangle such that its joins to the three corners trisect the triangle.

10. Find O within a triangle ABC so that the circum-circles of AOB, BOC, COA may be equal.

11. Through a given point, between two lines given in position, draw a line so that the segments of it between the point and the lines may have a given ratio.

12. Through a given point, within a given circle, draw a line so that the segments of it between the point and the circumference may have a given ratio.

13. Given a circle, and the positions of two lines; find P on the circle so that PX, PY, drawn parallel to given directions to meet the lines in X, Y, may have a given ratio.

14. From a given point, within a given angle, draw lines to meet the arms of the angle, so as to be in a given ratio and contain a given angle.

15. Through a given point P, within a given angle C, draw a line AB, to make with the arms of the given angle, a triangle ABC of given area.

ANALYSIS-Let CQ be drawn to meet round ABC in Q, so that СА. СВ,

AĈQ = BCP; then ▲3 ACQ, PCB are simr.; .. CQ. CP

=

which is known; and as also QAP PĈA, Q can be found. Cf. Problem 2.

16. Describe a circle to go through a fixed point, touch a fixed line, and have its centre in another fixed line.

NOTE-Let P be given pt., AB the line to be touched, and AC to contain the centre: then if BQ, I from P on AB, meets AC in Q, and QX is inflected to AP so that QX QB, PC || to XQ will meet AC in the reqd. centre.

=

17. Given three points; find a fourth, so that for every line through it, if perpendiculars are drawn from the three points on the line, the sum of two of them is equal to the third.

NOTE―The pt. reqd. is the mean centre of the three pts. when the mults. are each unity.

18. Given the three altitudes of a triangle, construct it.

19. If X, Y, Z are the points of contact of the in-circle with the sides respectively opposite the corners A, B, C of a triangle; find P, so that—

BPX = CPX, CPY = APY, and APZ = BPZ.

NOTE-Use vi. Addenda (17).

20. Construct a cyclic quadrilateral, the lengths of whose sides are given. 21. Draw a transversal to a given triangle, so that the segments of it intercepted between the sides (or sides produced) may have given lengths.

22. In a given triangle inscribe another of given species, one of whose sides shall go through a given point.

23. Given the direction of the base of a triangle, and the point at which it is touched by the in-circle; and given also the radius of the in-circle, and the difference of the other sides; find the Locus of the vertex.

24. Construct a triangle when given its vertical angle, the sum of the sides forming that angle, and the difference of the segments of the base made by the foot of the altitude.

25. Find a point in one side of a triangle such that the sum of parallels from it to the other two sides (terminated by them) may be equal to a given length. 26. O is a fixed point, OA a fixed direction; if a circle of fixed radius rolls along OA, and OP is drawn to touch it, and produced to Q, so that OP. PQ (radius)2, find the Locus of Q.

=

27. Given the sum of two sides of a triangle, an angle opposite either of these sides, and the radius of the in-circle; construct the triangle.

28. From a fixed point A any line is drawn to meet a fixed line in P; if AQ is drawn so that the angle PAQ, and the rectangle under AP, AQ are of given magnitude, find the Locus of Q.

29. Through fixed points A, B, outside a fixed circle, draw AXP, BYP, so that P, X, Y may be on the circle, and XY parallel to a given direction.

30. ABCD is a quadrilateral which varies subject to the following conditions: the corners A, C are fixed; the species of the triangle BCD is fixed; and the ratio of the rectangles under the opposite sides is fixed: find the Locus of either of the free corners.

NOTE-Use the construction of vi. Addenda (9), and the result of vi. Addenda (22).

31. Draw a parallel to one side of a triangle so that of the intercepts between it and that side

32. Given the lengths of the sides of a quadrilateral, and of the join of the mid points of one pair of opposite sides; construct the quadrilateral.

33. Draw the triangle of minimum perimeter, which has two corners, one on each of two fixed lines, and the third corner coincident with a fixed point. When is a solution impossible?

34. Show that the Problem-To inscribe a quadrilateral of minimum perimeter in a given quadrilateral is either indeterminate, or impossible.

35. Find the point in one side of a triangle the sum of whose distances from the other two sides is minimum.

When is there no minimum?

36. Find the point the sum of whose distances from the three sides of a triangle is minimum.

37. About a given triangle circumscribe the maximum equilateral triangle. 38. With the corners of a triangle as centres describe three circles to touch two and two.

39. Given the base of a triangle, and the length of the line drawn from one end of the base to cut the opposite side in a given ratio; find the Locus of the

vertex.

40. Inscribe a square in a triangle—

1o, by using proportions; and

2o, without using proportions.

41. About a given quadrilateral circumscribe a quadrilateral of given species. 42. In a given quadrilateral inscribe a quadrilateral of given species.

43. Given two intersecting circles, and a point in the area common to them, draw the line through the point which divides that common area into parts whose difference is maximum.

NOTE-Take MN the chd. of

(centre A) which is to AP at P; and let which is the image of A, with respect to MN, cut the other © in X: XP is the reqd. line.

the

=

44. Through A, one of the points of intersection of two circles, draw the double chord XAY, so that a . AX + b. AY c2; where a, b, c are given lengths. 45. Describe a circle so that the angles it subtends at three given points may be respectively equal to given angles.

46. Describe a circle so that the tangents to it from three given points may be respectively equal to given lengths.

47. Find the Locus of the point of contact of two variable circles, which touch two fixed circles and touch each other.

48. Find the Locus of the point from which tangents to two fixed circles are in a given ratio.

NOTE-Use General Addenda, iv. 2. Cor. (2).

49. OX, OY are fixed lines at right angles; and P is a fixed point in the bisector of the angle XOY: find a construction to give X so that the line XPY may be of a given length. (Pappus' Problem.)

=

NOTE-Drop PM ↓ to OX, and PN 1 to OY; and produce MP to L, so that PL given length: with centre N, and radius NL describe a O, meeting NP produced in H, K: then the Os on HP, KP as diams. will (when a solution is possible) by their intersections with OX, give four positions of X that solve the problem.