any Thus, the cross-ratio of a pencil is equal to that of transversal, and cross-ratios are unaltered by projection. Further, it can be shown that (ABCD) = (BADC) = (CDAB) = (DCBA), (ABCD).(ABDC)=1, (ABCD)+(ACBD)=1. The harmonic property of the complete quadrilateral follows. For (Fig. 85), (XYPQ) ̄r(UVMQ), and also ̄‚(VUMQ). Therefore (UV, MQ)=(VU, MQ) = − 1. If A (x1,y1, 21) and B(x2, y2, 22) are two fixed points, and P a variable point with coordinates respect to A and B. If Q is the point corresponding to λ the parameter μ, the cross-ratio (AB, PQ) The cross Ex. IV.] CROSS-RATIO 149 ratio of the two pairs of points corresponding to the parameters λ,' and u, u' is These results are the same as in euclidean geometry. EXAMPLES IV. 1. Prove that the actual Weierstrass line-coordinates of the absolute polar of (x, y, z) are (x/k, y/k, kz), and the actual pointcoordinates of the absolute pole of (§, ŋ, §) are (kέ, kŋ, Ĉ/k). 2. If the distance between the points (x1, Y1, 1), (X2, Y2, Z2) vanishes, prove that their join touches the absolute. 3. If (x1+ix2, Y1+iy2, z1+iz2) are the actual Weierstrass coordinates of a point (x1, y1, etc., being real numbers), prove that (x1, Y1, Z1) and (~2, Y2, Z2) are conjugate with regard to the absolute. 1 4. If (x1+ix2,...) (α1 + ia2,...) are the actual Weierstrass coordinates of two points at a real distance (x1, y1, etc., being real numbers), prove that, for all values of λ, (x1+λα1,...) and (x2+λɑ2,...) are conjugate with regard to the absolute. 5. If ds is the element of arc of a curve and dx, dy, dz the differentials of the Weierstrass coordinates, prove that ds2 = dx2 + dy2 + k2dz2. If r, are the polar coordinates, prove that ds2=dr2 + k2 sin2 = d✪2. k 6. ABCD is a skew quadrilateral, PQRS are points on the four sides AB, BC, CD, DA. Prove that if sin AP sin BQ sin CR sin DS = sin BP sin CQ sin DR sin AS, the four points PQRS lie in one plane. 7. 1, 2, 3, 4 are the vertices of a tetrahedron. A plane cuts each of the six edges. If the edge 12 is cut at A, and the ratio sin 1A/sin 24 is denoted by (12), prove that (12)(23)(34)(41) = 1. Conversely, if (12)(23)(34)(41)=1, prove that the points 12, 23, 34, 41 (i.e. the corresponding points on these edges) are coplanar. 8. If (12)(23)(34) (41)=1=(12)(24)(43)(31)=(13)(32) (24) (41), prove that either (i) the sets of points 12, 23, 31, etc., are collinear. or (ii) the lines (12, 34), (13, 24), (14, 23) are concurrent. 9. Four circles touch in succession, each one touching two others (the number of external contacts being even) show that the four points of contact lie on a circle, and that the four tangents at the points of contact touch a circle. 10. Four spheres touch in succession, each one touching two others (the number of external contacts being even); show that the four points of contact lie on a circle, and that the four tangent planes at the points of contact touch a sphere. Show further that, whatever the nature of the contacts, the four tangent planes pass through one point. 11. Five spheres touch in succession, each one touching two others (the number of external contacts being even); show that the five points of contact lie on a sphere, and that the five tangent planes at the points of contact touch a sphere. (Educ. Times (n.s.), xi. p. 57.) 12. D, E, F are the feet of the perpendiculars from a point ✪ on the sides of the triangle ABC. Prove that cos BD cos CE cos AF = cos CD cos AE cos BF. 13. ABC is a given triangle, and I is any line. P, Q, R are the feet of the perpendiculars from A, B, C on l. PP'1BC, QQʻICA, RR'LAB. Prove that PP', QQ', RR' meet in a point (the orthopole of l). 14. Prove that the locus of a point such that the ratio of the cosines of its distances from two fixed points is constant is a straight line. 15. If L, M, N; L1, M1, N1; etc., are the points of contact of the in- and e-scribed circles of the triangle ABC with the sides a, b, c, and 2s=a+b+c, prove the relations: 2 AM1=AN1=BN2=BL2=CL2=CM3=s, 16. Establish the reciprocal relations to those in Question 15 for the circumcircles. 17. Prove that the envelope of a line which makes with two fixed lines a triangle of constant perimeter is a circle. Prove also that the envelope is a circle if the excess of the sum of two sides over the third side is constant. What is the reciprocal theorem ? Ex. IV.] EXAMPLES 151 (In the following questions, 18-22, the geometry is hyperbolic. The formulae are analogous to well-known formulae in spherical trigonometry.) 18. If ka, kb, kc are the sides, and A, B, C the angles of a triangle, prove that 19. If r, r1, r2, rs are the radii of the in- and e-scribed circles of a 3 triangle ABC, prove that tanh r sinh s=tanh rı sinh (s – a)=tanh r, sinh (s - b) tanh rị tanh r, tanhrs=N/sinh (s – a) sinh (s – b) sinh (8 - c). 3 21. If R is the radius of the circumcircle of the triangle ABC, prove that 3 If D1, D2, D2 are the distances of the circumscribed equidistant-curves, prove that eoth R+tanh D、+tanh D,+tanh Dạ=2 cosh 8 sin 4/sinh a 1 3 3 coth R+tanh Dị - tanh D, - tanh Dạ=2 cosh (s – a) sin A/sinh a, etc. 23. Prove that, in the desmic configuration in § 19, the following sets of points are coplanar: áÃ1⁄2Ð ̧В1⁄2Ñ‚Ñ1⁄2‚ à ̧Ã¡В ̧В ̧уÑ1⁄2, and those obtained from these by cyclic permutation of ABC or of 234. Deduce that the configuration has the symbol 2 12 4 6 3 16 3 2, 4 3 24. If one pair of altitudes of a tetrahedron ABCD intersect, prove that the other pair will also intersect and if one altitude intersects two others, all four are concurrent. If these conditions are satisfied, prove that cos AB cos CD=cos AC cos BD=cos AD cos BC. 25. Prove that there is a circle which touches the in- and the e-scribed circles of a triangle. [In spherical geometry this is Hart's circle, and corresponds to the nine-point circle in ordinary geometry. See M'Clelland and Preston's Spherical Trigonometry, Chap. VI. Art. 88.] 26. Prove that there is a circle which touches the four circumcircles of a triangle. [In euclidean geometry the circumscribed equidistant-curves are three pairs of parallel lines and form a triangle A'B'C', of which A, B, C are the middle points of the sides. The circumcircle of ABC is the nine-point circle of A'B'C', and touches the inscribed circle of A'B'C'. That is, the last-named circle touches the four" circumcircles" of the triangle ABC.] |