= 11. Thrice the sum of the squares on the sides of any pentagon the sum of the squares on the diagonals together with four times the sum of the squares on the five straight lines joining, in order, the middle points of those diagonals. 12. If A, B be fixed points, and O any other point, the sum of the squares on OA and OB is least when O is the middle point of AB. 13. Prove II. 9, 10 by the following construction: On AD describe a rectangle A EFD whose sides AE, DF are each = AC or CB. According as D is in AB, or in AB produced, from DF, or DF produced, cut off FG = DB; and join EC, CG, GE. Show how these figures may be derived from those in the text. 14. If from the vertex of the right angle of a right-angled triangle a perpendicular be drawn to the hypotenuse, then (1) the square on this perpendicular is equal to the rectangle contained by the segments of the hypotenuse; (2) the square on either side is equal to the rectangle contained by the hypotenuse and the segment of it adjacent to that side. 15. The sum of the squares on two unequal straight lines is greater than twice the rectangle contained by the straight lines. 16. The sum of the squares on three unequal straight lines is greater than the sum of the rectangles contained by every two of the straight lines. 17. The square on the sum of three unequal straight lines is greater than three times the sum of the rectangles contained by every two of the straight lines. 18. The sum of the squares on the sides of a triangle is less than twice the sum of the rectangles contained by every two of the sides. 19. If one side of a triangle be greater than another, the median drawn to it is less than the median drawn to the other. 20. If a straight line AB be bisected in C, and divided internally at D and E, D being nearer the middle than E, then AD DB AE. EB + CD. DE + CE. ED. = 21. ABC is an isosceles triangle having each of the angles B and C = 2 A. BD is drawn 1 AC; prove AD2 + DC2 = 2 BD2. 22. Divide a given straight line internally so that the squares on the whole and on one of the segments may be double of the square on the other segment. 23. Given that AB is divided internally at H, and externally at H', in medial section, prove the following: (1) AH BH = (AH + BH) · (AH AH BH' = BH); = (BH' + AH') · (BH' – AH'). (2) AH. (AH – BH) (3) AB2 + BH2 (4) (AB + BH)2 = = 5 AH2; AB2 + BH'2 = 3 AH'2. (AB + BH')2 = 5 AH'2. (5) (AH - BH)2 = 3 BH2 AH2; (BH' - AH')2 = 3 AH'2 – BH'2. (6) (AH+BH)2 = 3 A H2 – BH2; (AH'+BH')2 = 3 BH'2 – AH'2. (7) (AB+ AH )2 = 8 AH 2 – 3 BH2; (AH' – AB)2=8 AH12 – 3 BH". (8) AB2 + AH2. = 4 AH2 – BH2; AB' + AH2 – 4 AH22 – BH'2. 24. In any triangle ABC, if BP, CQ be drawn 1 CA, BA, produced if necessary, then shall BC2 AB. BQ + AC.CP. 25. If from the hypotenuse of a right-angled triangle segments be cut off equal to the adjacent sides, the square of the middle segment thus formed twice the rectangle contained by the extreme segments. Show how this theorem may be used to find numbers expressing the sides of a right-angled triangle. (Leslie's Elements of Geometry, 1820, p. 315.) Loci. 1. Given a ▲ ABC; find the locus of the points the sum of the squares of whose distances from B and C, the ends of the base, is equal to the sum of the squares of the sides AB, AC. 2. Given a ▲ ABC; find the locus of the points the difference of the squares of whose distances from B and C, the ends of the base, is equal to the difference of the squares of the sides AB, AC. 3. Of the ▲ ABC, the base BC is given, and the sum of the sides AB, AC; find the locus of the point where the perpendicular from C to AC meets the bisector of the exterior vertical angle at A. 4. Of the ▲ ABC, the base BC is given, and the difference of the sides AB, AC; find the locus of the point where the perpendicular from C to AC meets the bisector of the interior vertical angle at A. 5. A variable chord of a given circle subtends a right angle at a fixed point; find the locus of the middle point of the chord. Examine the cases when the fixed point is inside the circle, outside the circle, and on the Oce 154 BOOK III. DEFINITIONS. 1. A circle is a plane figure contained by one line which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal. This point is called the centre of the circle, and the straight lines drawn from the centre to the circumference are called radii. COR. 1.-If a point be situated inside a circle, its distance from the centre is less than a radius; and if it be situated outside, its distance from the centre is greater than a radius. Fig. 1. Thus, in fig. 1, OP, the distance of the point P from the centre O, is less than the radius OA; in fig. 2, OP is greater than the radius OA. OA Fig. 2. P Ꭺ COR. 2.-Conversely, if the distance of a point from the centre of a circle be less than a radius, the point must be situated inside the circle; if its distance from the centre be greater than a radius, it must be situated outside the circle. COR. 3.-If the radii of two circles be equal, the circumferences are equal, and so are the circles themselves. This may be rendered evident by applying the one circle to the other, so that their centres shall coincide. Since the radii of the one circle are equal to those of the other, every point in the circum B E ference of the one circle will coincide with a point in the circumference of the other; therefore, the two circumferences COR. 4.-Conversely, if two circles be equal, their radii are equal, and also their circumferences. This may be proved indirectly, by supposing the radii unequal. COR. 5.-A circle is given in magnitude when the length of its radius is given, and a circle is given in position and magnitude when the position of its centre and the length of its radius are given. (Euclid's Data, Definitions 5 and 6.) COR. 6.-The two parts into which a diameter divides a circle are equal. This may be proved, like Cor. 3, by superposition. COR. 7.-The two parts into which a straight line not a diameter divides a circle are unequal. Thus if AB is not a diameter of the circle ABC, the two parts ACB and ADB into which AB divides the circle are unequal. For if a diameter AE be drawn, the part ACB is less than the semicircle ABE, and the part ADB is greater than the semicircle ADE. с B 2. Concentric circles are those which have a common centre. 3. A straight line is said to touch a circle, or to be a tangent to it, when it meets the circle, but being produced does not cut it. Thus BC is a tangent to the circle ADE. K B A о D E с 4. A straight line drawn from a point outside a circle, and cutting the circumference, is called a secant. Thus ECA and EBD are secants of the circle ABC. If the secant ECA were, like one of the hands of a watch, to revolve round E as a pivot, the points A and C would approach one another, and at D length coincide. When the points A and coincided, the secant would A E B have become a tangent. Hence a tangent to a circle may be defined to be a secant in its limiting position, or a secant which meets the circle in two coincident points. This way of regarding a tangent straight line may be applied also to a tangent circle. 5. Circles which meet but do not cut one another, are said to touch one another. Thus the circles ABC, ADE, which meet but do not intersect, are said to touch each other. In fig. 1, the circles are said to touch one another internally, although in strictness only one of them touches the other internally; in fig. 2, they are said to touch one another externally. 6. The points at which circles touch each other, or at which straight lines touch circles, are called points of contact. Thus in the figures to definitions 3 and 5, the points A are points of contact. |