11. If the sines of the angles A, B, C, of a triangle are in A.P, shew that 12. A regular polygon is inscribed in a circle, and the tangent of half the acute angle which a side subtends at the circumference = t; shew that a side of the figure the diameter of the circle : 2t: 1+ť. 13. Required the perpendicular from the vertex upon the base of a triangular pyramid, all the sides of which are equilateral triangles of given area. 14. A circle is inscribed in an equilateral triangle, an equilateral triangle in the circle, a circle in the latter triangle, and so on without limit; if r, r1, ", " ... be the radii of the 72, 73 circles, prove that r = r1 + r2+r2+ ...... 3 1 2 15. The sides of a plane triangle are as 3, 5, 6; compare the radii of the inscribed and circumscribed circles. 16. If from any point O within a triangle, three straight lines be drawn from the angles A, B, C, meeting the opposite sides in a, b, c, then will 17. In any right-angled plane triangle, twice the side of the inscribed square is an harmonic mean between the sides containing the right angle. 18. Prove that the area of a regular polygon inscribed in a circle is a geometrical mean between the areas of an inscribed and of a circumscribed polygon of half the number of sides; and that the area of a regular polygon circumscribed about a circle, is an harmonical mean between the areas of an inscribed one of the same number of sides, and of a circumscribed one of half that number. 19. If the side of a pentagon inscribed in a circle be 1, the radius is 5+√5 20. If R, r be the radii of the circumscribed and inscribed circles of a regular polygon of n sides, and R', r' the corresponding radii for a regular polygon of 2n sides, and of the same perimeter as the former, then 21. An indefinite area can be divided into no other regular figures than triangles, squares, and hexagons. 22. If A, B, C be the angular points of a triangle, a, b, c points in the sides respectively opposite to them, prove that if the lines Aa, Bb, Cc intersect in a point, then 24. If a quadrilateral is capable of having a circle inscribed in it, the sums of the opposite sides are equal to one another; and if, besides, it is capable of having one circumscribed about it, its area equals the square root of the continued product of the sides. 28. A lamp on the top of a pole 32 feet high is just seen by a man six feet in height, at a distance of 10 miles; find the earth's radius. 29. A ship, the height to the summit of the top mast of which from the water is 90 feet, is sailing directly towards an observer at the rate of 10 miles an hour; from the time of its first appearance in the offing till its arrival at the station of the observer is 1 hour 12 minutes; find approximately the earth's radius. 30. If O be the centre of the inscribed circle of a triangle, P, Q, R, the centres of the escribed circles, prove that OP2+ QR2 = OQ + RP2 = OR2 + PQ2. CONIC SECTIONS. THE PARABOLA. 1. THE locus of the vertices of all parabolas which have a common focus and a common tangent is a circle. 2. A circle touches the parabola at the vertex A. BA'C is that double ordinate of the axis which touches the circle at A', RPNP' any other ordinate of the axis, meeting the axis in N, and AB, produced if necessary, in R; shew that RP. RP' is proportional to AN. A'N. 3. A common tangent QP is drawn to a parabola and the circle described on its latus rectum as diameter; if S be the focus of the parabola, SQ, SP make equal angles with the latus rectum. 4. If SPG be an equilateral triangle, SP is equal to the latus rectum. 5. If the ordinate at P bisects the subnormal of P', then the semi-ordinate of P is equal to the normal at P'. 6. The portion of the tangent at the vertex intercepted between the vertex and any diameter is bisected by the tangent at the extremity of that diameter. 7. From P, Q extremities of any focal chord perpendiculars PL, QM are drawn to a given ordinate, shew that the sum of PL, QM, and PQ, is constant. 8. If a circle pass through the vertex and focus, cutting the tangent at the vertex in Q, the tangent to the circle at Q will touch the parabola. 9. A circle touches a parabola at A, cuts it at B, C, and the axis at E, BC meets the axis in D; if G be the middle point of DE, BG is a normal to the parabola at B. 10. The diameter of the circle passing through the extremities of the latus rectum and the vertex is five-fourths of the latus rectum. 11. If from the middle point of a chord two straight lines be drawn, one perpendicular to the chord, meeting the axis in G, the other to the axis meeting it in N, NG is constant. 12. The locus of the foot of the perpendicular on the normal from the focus is a parabola. 13. If SQ be perpendicular to the normal at P, QR to SP, PRAS. 14. If a parabola roll on an equal parabola, the vertices originally coinciding, the focus of the rolling traces out the directrix of the fixed parabola. 15. Draw a parabola to touch a given circle at a given point, so that its axis may touch the circle at another given point. 16. P is any point on a parabola whose vertex is A, and through the focus S the chord QSQ' is drawn parallel to AP, QM, Q'M' are drawn perpendicularly to the axis; shew that SM2 = AN. AM, and that AP= MM'. 17. If PM be the ordinate at P, T the intersection of the tangent at P with the axis, TP. TY TM. TS. = 18. If from any point in the tangent a straight line be drawn touching the parabola, the angle between this line and the line drawn to the focus from the same point is constant. 19. The portion of the directrix intercepted between the perpendiculars on it from the extremities of any focal chord subtends a right angle at the focus. 20. If the tangent at P meet the directrix in D, DSP is a right angle. |