Draw EG parallel to AB, and join DB, DG. Since the angle DEG is equal to the angle DGE; (1. 5.) and the angle GDF is equal to the angles DEG, DGE; (1. 32.) therefore the angle GDC is double of the angle DEG. But the angle BDC is equal to the angle BCD, (1. 5.) and the angle CEG is equal to the alternate angle ACE; (1. 29.) therefore the angle GDC is double of the angle CDB, add to these equals the angle CDB, therefore the whole angle GDB is treble of the angle CDB, but the angles GDB, CDB at the center D, are subtended by the arcs BF, BG, of which BG is equal to AE. Wherefore the circumference AE is treble of the circumference BF, and BF is one-third of AE. Hence may be solved the following problem: AE, BF are two arcs of a circle intercepted between a chord and a given diameter. Determine the position of the chord, so that one arc shall be triple of the other. PROPOSITION IV. THEOREM. AB, AC and ED are tangents to the circle CFB; at whatever point between C and B the tangent EFD is drawn, the three sides of the triangle AED are equal to twice AB or twice AC: also the angle subtended by the tangent EFD at the center of the circle, is a constant quantity. Take G the center of the circle, and join GB, GE, GF, GD, GC. Then EB is equal to EF, and DC to DF; (111. 37.) E F D A с therefore ED is equal to EB and DC; wherefore AD, AE, ED are equal to AB, AC; therefore AD, AE, ED are equal to twice AB, or twice AC; or the perimeter of the triangle AED is a constant quantity. Again, the angle EGF is half of the angle BGF, and the angle DGF is half of the angle CGF, therefore the angle DGE is half of the angle CGB, or the angle subtended by the tangent ED at G, is half of the angle contained between the two radii which meet the circle at the points where the two tangents AB, AC meet the circle. PROPOSITION V. PROBLEM. Given the base, the vertical angle, and the perpendicular in a plane triangle, to construct it. Upon the given base AB describe a segment of a circle containing an angle equal to the given angle. (111. 33.) D A B At the point B draw BC perpendicular to AB, and equal to the altitude of the triangle. (I. 11, 3.) Through C, draw CDE parallel to AB, and meeting the circumference in D and E. (1. 31.) Join DA, DB; also EA, EB; then EAB or DAB is the triangle required. It is also manifest, that if CDE touch the circle, there will be only one triangle which can be constructed on the base AB with the given altitude. PROPOSITION VI. THEOREM. If two chords of a circle intersect each other at right angles either within or without the circle, the sum of the squares described upon the four segments, is equal to the square described upon the diameter. Let the chords AB, CD intersect at right angles in E. A B F Draw the diameter AF, and join AC, AD, CF, DB. Then the angle ACF in a semicircle is a right angle, (111. 31.) and equal to the angle AED: also the angle ADC is equal to the angle AFC. (1.21.) Hence in the triangles ADE, AFC, there are two angles in the one respectively equal to two angles in the other, consequently, the third angle CAF is equal to the third angle DAB; therefore the arc DB is equal to the arc CF, (III. 26.) and therefore also the chord DB is equal to the chord CF. (III. 29.) Because AEC is a right-angled triangle, the squares on AE, EC are equal to the square on AC; (1. 47.) similarly, the squares on DE, EB are equal to the square on DB; therefore the squares on AE, EC, DE, EB, are equal to the squares on AC, DB; but DB was proved equal to FC, and the squares on AC, FC are equal to the square on AF, wherefore the squares on AE, EC, DE, EB, are equal to the square on AF, the diameter of the circle. When the chords meet without the circle, the property is proved in a similar manner. I. 7. THROUGH a given point within a circle, to draw a chord which shall be bisected in that point, and prove it to be the least. 8. To draw that diameter of a`given circle which shall pass at a given distance from a given point. 9. Find the locus of the middle points of any system of parallel chords in a circle. 10. The two straight lines which join the opposite extremities of two parallel chords, intersect in a point in that diameter which is perpendicular to the chords. 11. The straight lines joining towards the same parts, the extremities of any two lines in a circle equally distant from the center, are parallel to each other. 12. A, B, C, A', B', C' are points on the circumference of a circle; if the lines AB, AC be respectively parallel to A'B', A'C', shew that BC' is parallel to B'C. 13. Two chords of a circle being given in position and magnitude, describe the circle. 14. Two circles are drawn, one lying within the other; prove that no chord to the outer circle can be bisected in the point in which it touches the inner, unless the circles are concentric, or the chord be perpendicular to the common diameter. If the circles have the same center, shew that every chord which touches the inner circle is bisected in the point of contact. 15. Draw a chord in a circle, so that it may be double of its perpendicular distance from the center. 16. The arcs intercepted between any two parallel chords in a circle are equal. 17. If any point P be taken in the plane of a circle, and PA, PB, PC,..be drawn to any number of points A, B, C,..situated symmetrically in the circumference, the sum of PA, PB,..is least when P is at the center of the circle. II. 18. The sum of the arcs subtending the vertical angles made by any two chords that intersect, is the same, as long as the angle of intersection is the same. 19. From a point without a circle two straight lines are drawn cutting the convex and concave circumferences, and also respectively parallel to two radii of the circle. Prove that the difference of the concave and convex arcs intercepted by the cutting lines, is equal to twice the arc intercepted by the radii. 20. In a circle with center O, any two chords, AB, CD are drawn cutting in E, and OA, OB, OC, OD are joined; prove that the angles AOC BOD=2.AEC, and AOD + BOC=2.AED. 21. If from any point without a circle, lines be drawn cutting the circle and making equal angles with the longest line, they will cut off equal segments. 22. If the corresponding extremities of two intersecting chords of a circle be joined, the triangles thus formed will be equiangular. 23. Through a given point within or without a circle, it is required to draw a straight line cutting off a segment containing a given angle. 24. If on two lines containing an angle, segments of circles be described containing angles equal to it, the lines produced will touch the segments. 25. Any segment of a circle being described on the base of a triangle; to describe on the other sides segments similar to that on the base. 26. If an arc of a circle be divided into three equal parts by three straight lines drawn from one extremity of the arc, the angle contained by two of the straight lines is bisected by the third. 27. If the chord of a given circular segment be produced to a fixed point, describe upon it when so produced a segment of a circle which shall be similar to the given segment, and shew that the two segments have a common tangent. 28. If AD, CE be drawn perpendicular to the sides BC, AB of the triangle ABC, and DE be joined, prove that the angles ADE, and ACE are equal to each other. 29. If from any point in a circular arc, perpendiculars be let fall on its bounding radii, the distance of their feet is invariable. III. 30. If both tangents be drawn, (fig. Euc. III. 17.) and the points of contact joined by a straight line which cuts EA in H, and on HA as diameter a circle be described, the lines drawn through E to touch this circle will meet it on the circumference of the given circle. 31. Draw, (1) perpendicular, (2) parallel to a given line, a line touching a given circle. 32. If two straight lines intersect, the centers of all circles that can be inscribed between them, lie in two lines at right angles to each other. 33. Draw two tangents to a given circle, which shall contain an angle equal to a given rectilineal angle. 34. Describe a circle with a given radius touching a given line, and so that the tangents drawn to it from two given points in this line may be parallel, and shew that if the radius vary, the locus of the centers of the circles so described is a circle. 35. Determine the distance of a point from the center of a given circle, so that if tangents be drawn from it to the circle, the concave part of the circumference may be double of the convex. 36. In a chord of a circle produced, it is required to find a point, from which if a straight line be drawn touching the circle, the line so drawn shall be equal to a given straight line. 37. Find a point without a given circle, such that the sum of the two lines drawn from it touching the circle. shall be equal to the line drawn from it through the center to meet the circle. 38. If from a point without a circle two tangents be drawn; the straight line which joins the points of contact will be bisected at right angles by a line drawn from the center to the point without the circle. 39. If tangents be drawn at the extremities of any two diameters of a circle, and produced to intersect one another; the straight lines joining the opposite points of intersection will both pass through the center. 40. If from any point without a circle two lines be drawn touching the circle, and from the extremities of any diameter, lines be drawn to the point of contact cutting each other within the circle, the line drawn from the points without the circle to the point of intersection, shall be perpendicular to the diameter. 41. If any chord of a circle be produced equally both ways, and tangents to the circle be drawn on opposite sides of it from its extremities, the line joining the points of contact bisects the given chord. 42. AB is a chord, and AD is a tangent to a circle at A. DPQ any secant parallel to AB meeting the circle in P and Q. Shew that the triangle PAD is equiangular with the triangle QAB. 43. If from any point in the circumference of a circle a chord and tangent be drawn, the perpendiculars dropped upon them from the middle point of the subtended arc, are equal to one another. IV. 44. In a given straight line to find a point at which two other straight lines being drawn to two given points, shall contain a right angle. Shew that if the distance between the two given points be greater than the sum of their distances from the given line, there will be two such points; if equal, there may be only one; if less, the problem may be impossible. 45. Find the point in a given straight line at which the tangents to a given circle will contain the greatest angle. 46. Of all straight lines which can be drawn from two given points to meet in the convex circumference of a given circle, the sum of those two will be the least, which make equal angles with the tangent at the point of concourse. 47. DF is a straight line touching a circle, and terminated by AD, BF, the tangents at the extremities of the diameter AB, shew that the angle which DF subtends at the center is a right angle. 48. If tangents Am, Bn be drawn at the extremities of the diameter of a semicircle, and any line in mPn crossing them and touching the circle in P, and if AN, BM be joined intersecting in O and cutting the semicircle in E and F; shew that O, P, and the point of intersection of the tangents at E and F, are in the same straight line. 49. If from a point P without a circle, any straight line' be drawn cutting the circumference in A and B, shew that the straight lines joining the points A and B with the bisection of the chord of contact of the tangents from P, make equal angles with that chord. |