ST JOHN'S COLLEGE. DEC. 1837. (No. VIII.) 1. IF two triangles which have two sides of the one proportional to two sides of the other be joined at one angle so as to have their homologous sides parallel to one another; the remaining sides shall be in a straight line. 2. If a solid angle be contained by three plane angles, any two of them are greater than the third. 3. If from any point in the diagonal of a parallelogram, straight lines be drawn to the angles, the parallelogram will be divided into two pairs of equal triangles. 4. Shew how to find the focus of a traced conic section. 5. From three given centres describe three circles touching one another. 6. SY, HZ are perpendiculars from the foci on the tangent at P to an ellipse whose centre is C; SP, HP cut CY, CZ in Q, R; shew that CQPR is a parallelogram. 7. Let the two circles, radii R, r, which touch first, the three sides of a triangle ABC, and secondly one side BC and the other two produced, touch AB in D1, D2, AC in E1, E; shew that BD, . BD, = CE1. CE2 = Rr. 8. The side of an equilateral hexagon inscribed in an ellipse, eccentricity e, with two sides parallel to the axis major side of one inscribed in the circle on the axis-major :: 4 - 2e2: 4 - e2. 9. : If one of the co-ordinates of the centre of the curve ay2 + bxy + c x2 + dy + ex + ƒ = 0, assume the form that the equation becomes 0 and explain the meaning of it. 10. AP is a parabola, vertex A, focus S; T the point where the axis intersects the directrix; join PT and produce it to meet the latus rectum in N; draw SPQ to meet NQ, which is parallel to ST in Q; and shew that the locus of Q is a circle. 11. If O be a point in the directrix of a parabola; and OA = a, OB OB b, tangents at 4 and B; shew that the equation to the parabola referred to OA, OB as axes, assumes 13. If in (11) any tangent to the parabola cut OA, OB in P and Q; shew that where OP or OQ is considered as negative, if P or Q lies in AO or BO produced backwards. 14. Two parallel planes revolve in their own planes about fixed points A, B, in the same direction with equal angular velocities; shew that the curve traced upon the first by a pencil P fixed perpendicular to the plane of the second is a circle: or if the planes revolve in opposite directions the equation to the curve is where A is the origin, BP = a, AB = c, and the prime radius is the line originally in the position AB. 15. ABCD is any quadrilateral. Bisect AC, BD in E and F: EF is the locus of the centres of all the inscribed ellipses. 16. Shew that all lines drawn from an external point to touch a sphere are equal to one another; and thence prove that if a tetrahedron can have a sphere inscribed in it, touching its six edges, the sum of every two opposite edges is the same. SOLUTIONS TO (No. VIII.) 1. EUCLID, Prop. 32. Book vi. 2. Euclid, Prop. 20. Book x1. 3. Let ABCD (fig. 64) be the parallelogram whose diagonal is AC; E any point in it; join DE, EB; then since AACD = AABC, the perpendicular from D on AC equal the perpendicular from B on AC; hence the altitudes of the triangles ADE, ABE are equal; and they are upon the same base, therefore AADE = AABE. 4. Similarly Find C the centre of the ellipse (fig. 65) by joining the points of bisection of two parallel chords; take any point D in the curve, and with centre C and radius CD describe a circle cutting the ellipse in the four points D, E, F, G; through C draw AA', BB' parallel to DE, EF respectively; these will be the two axes; and with centre B and radius = AC describe a circle cutting AA' in S, H; these will be the two foci required. 5. Let A, B, C (fig. 66) be the three given centres; find O the centre of the circle inscribed in the triangle ABC; draw Oa, Ob, Oc perpendicular to the three sides BC, AC, AB respectively; then Ab Ac, Cb Ca, Ba - Be; and the three circles described with centres A, B, C and radii Ac, Bc, Ca, respectively, will touch one another in the points a, b, c. 6. = Produce HP, SY (fig. 67) to meet in V; then since <SPY = ¿YPV, SY YV, and HC CS; therefore HP is parallel to CY; similarly SP is parallel to CZ; or CQPR is a parallelogram. 7. BD1 = S-b; BD2 = S-c; CE, S-c; CE2=S-b; 2 .. BD1.BD,= (S′ − b) (S′ − c) = CE1 . CE2 ; 8. Let APQa (fig. 68) be half the hexagon, Aa being one of its diagonals; then if x, y be the co-ordinates of P measured from the centre C, PQ = 2∞, AP = 2x; and (a − x)2 + y2 = AP2 = 4x2, , or 3x2+2ax − a2 = (1 − e2) (a2 -- x2) ; 2 - e2 .. (4 and 2x = · e2) x2 + 2 ax = (2 - e2) a2; or ∞ = x a; and the side of the hexagon inscribed in the circle on the axis major = a; therefore the side of the hexagon inscribed in the ellipse: the side of the hexagon inscribed in the circle on the axis-major :: 42e: 4 — e3. 9. Let a, ẞ be the co-ordinates of the centre; transform the origin to that point by making x = x′ + a, y = y' + ß ; a (y' + ß)2 + b (x' + a) (y' + ß) + c (x' + a)2 In this case the equation to the curve becomes (2 ay + bx)2 + 2d (2ay + bx) + 4aƒ = 0, which represents the equations to two parallel straight lines; |