7. PQ is a fixed chord in a circle, and PX, QY any two parallel chords through P and Q; shew that XY touches a fixed concentric circle. 8. Two equal circles intersect at A and B; and from C, any point on the circumference of one of them, a perpendicular is drawn to AB, meeting the other circle at O and O'; shew that either O or O' is the orthocentre of the triangle ABC. Distinguish between the two cases. 9. Three equal circles pass through the same point A, and their other points of intersection are B, C, D : shew that of the four points A, B, C, D, each is the orthocentre of the triangle formed by joining the other three. 10. From a given point without a circle draw a straight line to the concave circumference so as to be bisected by the convex circumference. When is this problem impossible ? 11. Draw a straight line cutting two concentric circles so that the chord intercepted by the circumference of the greater circle may be double of the chord intercepted by the less. 12. ABC is a triangle inscribed in a circle, and A', B', C'are the middle points of the arcs subtended by the sides (remote from the opposite vertices): find the relation between the angles of the two triangles ABC, A'B'C'; and prove that the pedal triangle of A'B'C' is equiangular to the triangle ABC. 13. The opposite sides of a quadrilateral inscribed in a circle are produced to meet: shew that the bisectors of the two angles so formed are perpendicular to one another. 14. If a quadrilateral can have one circle inscribed in it, and another circunscribed about it; shew that the straight lines joining the opposite points of contact of the inscribed circle are perpendicular to one another. 15. Given the base of a triangle and the sum of the remaining sides; find the locus of the foot of the perpendicular from one extremity of the base on the bisector of the exterior vertical angle. 16. Two circles touch each other at C, and straight lines are drawn through C at right angles to one another, meeting the circles at P, P' and Q, Q' respectively: if the straight line which joins the centres is terminated by the circumferences at A and A', shew that P'p2 + Q'Q2=A'A?. 17. Two circles cut one another orthogonally at A and B; p is any point on the arc of one circle intercepted by the other, and PA, PB are produced to meet the circumference of the second circle at C and D: shew that CD is a diameter. 18. ABC is a triangle, and from any point P perpendiculars PD, PE, PF are drawn to the sides : if S1, S2, S, are the centres of the circles circumscribed about the triangles ÉPF, FPD, DPE, shew that the triangle S, S, Sz is equiangular to the triangle ABC, and that the sides of the one are respectively half of the sides of the other. 19. Two tangents PA, PB are drawn from an external point P to a given circle, and C is the middle point of the chord of contact AB; if XY is any chord through P, shew that AB bisects the angle XCY. 20. Given the sum of two straight lines and the rectangle contained by them (equal to a given square) : find the lines. 21. Given the sum of the squares on two straight lines and the rectangle contained by them : find the lines. 22. Given the sum of two straight lines and the sum of the squares on them : find the lines. 23. Given the difference between two straight lines, and the rectangle contained by them : find the lines. 24. Given the sum or difference of two straight lines and the difference of their squares : find the lines. 25. ABC is a triangle, and the internal and external bisectors of the angle A meet BC, and BC produced, at P and P': if O is the middle point of PP', shew that OA is a tangent to the circle circum. scribed about the triangle ABC. 26. ABC is triangle, and from P, any point on the circum. ference of the circle circumscribed about it, perpendiculars are drawn to the sides BC, CA, AB meeting the circle again in A', B', C'; prove that (i) the triangle A'B'C' is identically equal to the triangle ABC. (ii) AA', BB', CC' are parallel. 27. Two equal circles intersect at fixed points A and B, and from any point in AB a perpendicular is drawn to meet the circumferences on the same side of AB at P and Q: shew that PQ is of constant length. 28. The straight lines which join the vertices of a triangle to the centre of its circumscribed circle, are perpendicular respectively to the sides of the pedal triangle. 29. P is any point on the circumference of a circle circumscribed about a triangle ABC; and perpendiculars PD, PE are drawn from P to the sides BC, CA. Find the locus of the centre of the circle circumscribed about the triangle PDE. 30. P is any point on the circumference of a circle circumscribed about a triangle ABC: shew that the angle between Simson's Line for the point P and the side BC is equal to the angle between AP and the diameter of the circumscribed circle through A. 31. Shew that the circles circumscribed about the four triangles formed by two pairs of intersecting straight lines meet in a point. 32. Shew that the orthocentres of the four triangles formed by two pairs of intersecting straight lines are collinear. ON THE CONSTRUCTION OF TRIANGLES. 33. Given the vertical angle, one of the sides containing it, and the length of the perpendicular from the vertex on the base : construct the triangle. 34. Given the feet of the perpendiculars drawn from the vertices on the opposite sides : construct the triangle. 35. Given the base, the altitude, and the radius of the circumscribed circle: construct the triangle. 36. Given the base, the vertical angle, and the sum of the squares on the sides containing the vertical angle: construct the triangle. 37. Given the base, the altitude and the sum of the squares on the sides containing the vertical angle: construct the triangle. 38. Given the base, the vertical angle, and the difference of the squares on the sides containing the vertical angle: construct the triangle. 39. Given the vertical angle, and the lengths of the two medians drawn from the extremities of the base : construct the triangle. 40. Given the base, the vertical angle, and the difference of the angles at the base : construct the triangle. 41. Given the base, and the position of the bisector of the vertical angle : construct the triangle. 42. Given the base, the vertical angle, and the length of the bisector of the vertical angle: construct the triangle. 43. Given the perpendicular from the vertex on the base, the bisector of the vertical angle, and the median which bisects the base : construct the triangle. 44. Given the bisector of the vertical angle, the median bisecting the base, and the difference of the angles at the base : construct the triangle. BOOK IV. Book IV. consists entirely of problems, dealing with various rectilineal figures in relation to the circles which pass through their angular points, or are touched by their sides. DEFINITIONS. 1. A Polygon is a rectilineal figure bounded by more than four sides. A Polygon of five sides is called a Pentagon, six sides Hexagon, seven sides Heptagon, eight sides Octagon, ten sides Decagon, twelve sides Dodecagon, fifteen sides Quindecagon. 2. A Polygon is Regular when all its sides are equal, and all its angles are equal. 3. A rectilineal figure is said to be inscribed in a circle, when all its angular points are on the circumference of the circle; and a circle is said to be circumscribed about a rectilineal figure, when the circumference of the circle passes through all the angular points of the figure. 4. A circle is said to be inscribed in a rectilineal figure, when the circumference of thecircle is touched by each side of the figure; and a rectilineal figure is said to be circumscribed about a circle, when each side of the figure is a tangent to the circle. 5. A straight line is said to be placed in a circle, when its extremities are on the circumference of the circle. C PROPOSITION 1. PROBLEM. In a given circle to place a chord equal to a given straight line, which is not greater than the diameter of the circle. Let ABC be the given circle, and D the given straight line not greater than the diameter of the circle. It is required to place in the O ABC a chord equal to D. Then if CBD, the thing required is done. From CB cut off CE equal to D: and with centre C, and radius CE, describe the O AEF, cutting the given circle at A. Join CA. 1. 3. Proof. For CA=CE, being radii of the O AEF; Constr. .. CA=D. Q.E.F. EXERCISES. 1. In a given circle place a chord of given length so as to pass through a given point (i) without, (ii) within the circle. When is this problem impossible? 2. In a given circle place a chord of given length so that it may be parallel to a given straight line. |