28. ABC is a triangle, O is its orthocentre, and AK a diameter of the circumscribed circle: shew that BOCK is a parallelogram. 29. The orthocentre of a triangle is joined to the middle point of the base, and the joining line is produced to meet the circumscribed circle: prove that it will meet it at the same point as the diameter which passes through the vertex. 30. The perpendicular from the vertex of a triangle on the base, and the straight line joining the orthocentre to the middle point of the base, are produced to meet the circumscribed circle at P and Q: shew that PQ is parallel to the base. 31. The distance of each vertex of a triangle from the orthocentre is double of the perpendicular drawn from the centre of the circumscribed circle on the opposite side. 32. Three circles are described each passing through the orthocentre of a triangle and two of its vertices: shew that the triangle formed by joining their centres is equal in all respects to the original triangle. 33. ABC is a triangle inscribed in a circle, and the bisectors of its angles which intersect at O are produced to meet the circumference in PQR: shew that O is the orthocentre of the triangle PQR. 34. Construct a triangle, having given a vertex, the orthocentre, and the centre of the circumscribed circle. LOCI. 35. Given the base and vertical angle of a triangle, find the locus of its orthocentre. Let BC be the given base, and X the given angle; and let BAC be any triangle on the base BC, having its vertical A equal to the X. Draw the perps BE, CF, intersecting at the orthocentre O. E It is required to find the locus of O. Since the SOFA, OEA are rt. angles, the points O, F, A, E are concyclic; the FOE is the supplement of the LA: III. 22. ..the vert. opp. Z BOC is the supplement of the L A. But the A is constant, being always equal to the X; its supplement is constant ; that is, the ▲ BOC has a fixed base, and constant vertical angle; hence the locus of its vertex O is the arc of a segment of which BC [See Corollary p. 201.] is the chord. 36. Given the base and vertical angle of a triangle, find the locus of the intersection of the bisectors of its angles. Let BAC be any triangle on the given base BC, having its vertical angle equal to the given X; and let Al, BI, CI be the bisectors of its angles. [See Ex. 2, p. 111.] It is required to find the locus of the point 1. Denote the angles of the A ABC by A, B, C; and let the L BIC be denoted by I. Then from the ▲ BIC, (i) and from the ▲ ABC, or, I+B+C=two rt. angles, A B I. 32. I. 32. (ii) so that A+B+C=one rt. angle, .., taking the differences of the equals in (i) and (ii), =one rt. angle + A. But A is constant, being always equal to the ▲ X; I is constant: .. the locus of I is the arc of a segment on the fixed chord BC. 37. Given the base and vertical angle of a triangle, find the locus of the centroid, that is, the intersection of the medians. Let BAC be any triangle on the given base BC, having its vertical angle equal to the given angle S; let the medians AX, BY, CZ intersect at the centroid G. [See Ex. 4, p. 113.] It is required to find the locus of the point G. Through G draw GP, GQ par1 to AB and AC respectively. Then ZG is a third part of ZC; Ex. 4, p. 113, B P and since GP is par1 to ZB, Similarly QC is a third part of BC; .. P and Q are fixed points. Ex. 19, p. 107. Constr. I. 29. Now since PG, GQ are par1 respectively to BA, AC, the locus of G is the arc of a segment on the fixed chord PQ. NOTE. In this problem the points A and G move on the arcs of similar segments. 38. Given the base and the vertical angle of a triangle; find the locus of the intersection of the bisectors of the exterior base angles. 39. Through the extremities of a given straight line AB any two parallel straight lines AP, BQ are drawn; find the locus of the intersection of the bisectors of the angles PÁB, QBA. 40. Find the locus of the middle points of chords of a circle drawn through a fixed point. Distinguish between the cases when the given point is within, on, or without the circumference. 41. Find the locus of the points of contact of tangents drawn from a fixed point to a system of concentric circles. 42. Find the locus of the intersection of straight lines which pass through two fixed points on a circle and intercept on its circumference an arc of constant length. 43. A and B are two fixed points on the circumference of a circle, and PQ is any diameter: find the locus of the intersection of PA and QB. 44. BAC is any triangle described on the fixed base BC and having a constant vertical angle; and BA is produced to P, so that BP is equal to the sum of the sides containing the vertical angle: find the locus of P. 45. AB is a fixed chord of a circle, and AC is a moveable chord passing through A: if the parallelogram CB is completed, find the locus of the intersection of its diagonals. 46. A straight rod PQ slides between two rulers placed at right angles to one another, and from its extremities PX, QX are drawn perpendicular to the rulers: find the locus of X. 47. Two circles whose centres are C and D, intersect at A and B: through A, any straight line PAQ is drawn terminated by the circumferences; and PC, QD intersect at X: find the locus of X, and shew that it passes through B. [Ex. 9, p. 234.] 48. Two circles intersect at A and B, and through P, any point on the circumference of one of them, two straight lines PA, PB are drawn, and produced if necessary, to cut the other circle at X and Y: find the locus of the intersection of AY and BX. 49. Two circles intersect at A and B; HAK is a fixed straight line drawn through A and terminated by the circumferences, and PAQ is any other straight line similarly drawn: find the locus of the intersection of HP and QK. 50. Two segments of circles are on the same chord AB and on the same side of it; and P and Q are any points one on each arc: find the locus of the intersection of the bisectors of the angles PAQ, PBQ. 51. Two circles intersect at A and B; and through A any straight line PAQ is drawn terminated by the circumferences: find the locus of the middle point of PQ. MISCELLANEOUS EXAMPLES ON ANGLES IN A CIRCLE. are 52. ABC is a triangle, and circles are drawn through B, C, cutting the sides in P, Q, P', Q', ... shew that PQ, P'Q' parallel to one another and to the tangent drawn at A to the circle circumscribed about the triangle. 53. Two circles intersect at B and C, and from any point A, on the circumference of one of them, AB, AC are drawn, and produced if necessary, to meet the other at D and E: shew that DE is parallel to the tangent at A. 54. A secant PAB and a tangent PT are drawn to a circle from an external point P; and the bisector of the angle ATB meets AB at C: shew that PC is equal to PT. 55. From a point A on the circumference of a circle two chords AB, AC are drawn, and also the diameter AF: if AB, AC are produced to meet the tangent at F in D and E, shew that the triangles ABC, AED are equiangular to one another. 56. O is any point within a triangle ABC, and OD, OE, OF are drawn perpendicular to BC, CA, AB respectively: shew that the angle BOC is equal to the sum of the angles BAC, EDF. 57. If two tangents are drawn to a circle from an external point, shew that they contain an angle equal to the difference of the angles in the segments cut off by the chord of contact. 58. Two circles intersect, and through a point of section a straight line is drawn bisecting the angle between the diameters through that point: shew that this straight line cuts off similar seginents from the two circles. 59. Two equal circles intersect at A and B; and from centre A, with any radius less than AB a third circle is described cutting the given circles on the same side of AB at C and D: shew that the points B, C, D are collinear. 60. ABC and A'B'C' are two triangles inscribed in a circle, so that AB, AC are respectively parallel to A'B', A'C': shew that BC' is parallel to B'C. 61. Two circles intersect at A and B, and through A two straight lines HAK, PAQ are drawn terminated by the circumferences: if HP and KQ intersect at X, shew that the points H, B, K, X are concyclic. 62. Describe a circle touching a given straight line at a given point, so that tangents drawn to it from two fixed points in the given line may be parallel. [See Ex. 10, p. 197.] 63. C is the centre of a circle, and CA, CB two fixed radii: if from any point P on the arc AB perpendiculars PX, PY are drawn to CA and CB, shew that the distance XY is constant. 64. AB is a chord of a circle, and P any point in its circumference; PM is drawn perpendicular to AB, and AN is drawn perpendicular to the tangent at P: shew that MN is parallel to PB. 65. P is any point on the circumference of a circle of which AB is a fixed diameter, and PN is drawn perpendicular to AB; on AN and BN as diameters circles are described, which are cut by AP, BP at X and Y shew that XY is a common tangent to these circles. : 66. Upon the same chord and on the same side of it three segments of circles are described containing respectively a given angle, its supplement and a right angle: shew that the intercept made by the two former segments upon any straight line drawn through an extremity of the given chord is bisected by the latter segment. 67. Two straight lines of indefinite length touch a given circle, and any chord is drawn so as to be bisected by the chord of contact: if the former chord is produced, shew that the intercepts between the circumference and the tangents are equal. 68. Two circles intersect one another: through one of the points of section draw a straight line of given length terminated by the circumferences. 69. On the three sides of any triangle equilateral triangles are described remote from the given triangle: shew that the circles described about them intersect at a point. 70. On BC, CA, AB the sides of a triangle ABC, any points P, Q, R are taken; shew that the circles described about the triangles AQR, BRP, CPQ meet in a point. 71. Find a point within a triangle at which the sides subtend equal angles. 72. Describe an equilateral triangle so that its sides may pass through three given points. 73. Describe a triangle equal in all respects to a given triangle, and having its sides passing through three given points. |