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24. In any triangle the sum of the squares on two sides is equal to twice the square on half the third side together with twice the square on the median which bisects the third side.
Let ABC be a triangle, and AP the median bisecting the side BC. Then shall AB2+AC2=2BP2+2 AP2.
Draw AQ perp. to BC.
Consider the case in which AQ falls within the triangle, but does not coincide with AP.
Now of the angles APB, APC, one must be obtuse, and the other acute let APB be obtuse.
APB, AB2=BP2+AP2+2BP. PQ,
Then in the
Also in the
AC2=CP2+AP2 - 2CP. PQ.
But CP BP,
.. CP2=BP2; and the rect. BP, PQ=the rect. CP, PQ,
Hence adding the above results,
Q. E. D.
The student will have no difficulty in adapting this proof to the cases in which AQ falls without the triangle, or coincides with AP.
25. The sum of the squares on the sides of a parallelogram is equal to the sum of the squares on the diagonals.
26. In any quadrilateral the squares on the diagonals are together equal to twice the sum of the squares on the straight lines joining the middle points of opposite sides. [See Ex. 9, p. 105.]
27. If from any point within a rectangle straight lines are drawn to the angular points, the sum of the squares on one pair of the lines drawn to opposite angles is equal to the sum of the squares on the other pair.
28. The sum of the squares on the sides of a quadrilateral is greater than the sum of the squares on its diagonals by four times the square on the straight line which joins the middle points of the diagonals.
29. O is the middle point of a given straight line AB, and from O as centre, any circle is described: if P be any point on its circumference, shew that the sum of the squares on AP, BP is constant.
30. Given the base of a triangle, and the sum of the squares on the sides forming the vertical angle; find the locus of the vertex.
31. ABC is an isosceles triangle in which AB and AC are equal. AB is produced beyond the base to D, so that BD is equal to AB. Shew that the square on CD is equal to the square on AB together with twice the square on BC.
32. In a right-angled triangle the sum of the squares on the straight lines drawn from the right angle to the points of trisection of the hypotenuse is equal to five times the square on the line between the points of trisection.
33. Three times the sum of the squares on the sides of a triangle is equal to four times the sum of the squares on the medians. 34. ABC is a triangle, and O the point of intersection of its medians shew that
AB2+ BC2 + CA2 = 3 (OA2 + OB2+OC2).
35. ABCD is a quadrilateral, and X the middle point of the straight line joining the bisections of the diagonals; with X as centre any circle is described, and P is any point upon this circle: shew that PA2+ PB2+ PC2+PD2 is constant, being equal to
XA2+ XB2+XC2 + XD2 + 4XP2.
36. The squares on the diagonals of a trapezium are together equal to the sum of the squares on its two oblique sides, with twice the rectangle contained by its parallel sides.
37. Construct a rectangle equal to the difference of two squares.
38. Divide a given straight line into two parts so that the rectangle contained by them may be equal to the square described on a given straight line which is less than half the straight line to be divided.
39. Given a square and one side of a rectangle which is equal to the square, find the other side.
40. Produce a given straight line so that the rectangle contained by the whole line thus produced and the part produced, may be equal to the square on another given line.
41. Produce a given straight line so that the rectangle contained by the whole line thus produced and the given line shall be equal to the square on the part produced.
42. Divide a straight line AB into two parts at C, such that the rectangle contained by BC and another line X may be equal to the square on AC.
Book III. deals with the properties of Circles.
For convenience of reference the following definitions are repeated from Book I.
1. Def. 15. A circle is a plane figure bounded 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 to one another: this point is called the centre of the circle.
NOTE. Circles which have the same centre are said to be concentric.
I. Def. 16. A radius of a circle is a straight line drawn from the centre to the circumference.
I. Def. 17. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
I. Def. 18. A semicircle is the figure bounded by a diameter of a circle and the part of the circumference cut off by the diameter.
NOTE. From these definitions we draw the following inferences:
(i) The distance of a point from the centre of a circle is less than the radius, if the point is within the circumference: and the distance of a point from the centre is greater than the radius, if the point is without the circumference.
(ii) A point is within a circle if its distance from the centre is less than the radius: and a point is without a circle if its distance from the centre is greater than the radius.
(iii) Circles of equal radius are equal in all respects; that is to say, their areas and circumferences are equal.
(iv) A circle is divided by any diameter into two parts which are equal in all respects.
DEFINITIONS TO BOOK III.
1. An arc of a circle is any part of the circumference.
2. A chord of a circle is the straight line which joins any two points on the circumference.
NOTE. From these definitions it may be seen that a chord of a circle, which does not pass through the centre, divides the circumference into two unequal arcs; of these, the greater is called the major arc, and the less the minor arc. Thus the major arc is greater, and the minor arc less than the semi-circumference.
The major and minor arcs, into which a circumference is divided by a chord, are said to be conjugate to one another.
3. Chords of a circle are said to be equidistant from the centre, when the perpendiculars drawn to them from the centre are equal:
and one chord is said to be further from the centre than another, when the perpendicular drawn to it from the centre is greater than the perpendicular drawn to the other.
A secant of a circle is a straight line of indefinite length, which cuts the circumference in two points.
5. A tangent to a circle is a straight line which meets the circumference, but being produced, does not cut it. Such a line is said to touch the circle at a point; and the point is called the point of contact.
NOTE. If a secant, which cuts a circle at the points P and Q, gradually changes its position in such a way that P remains fixed, the point will ultimately approach the fixed point P, until at length these points may be made to coincide. When the straight line PQ reaches this limiting position, it becomes the tangent to the circle at the point P.
Hence a tangent may be defined as a straight line which passes through two coincident points on the circumference.
6. Circles are said to touch one another when they meet, but do not cut one another.
When each of the circles which meet is outside the other, they are said to touch one another externally, or to have external contact: when one of the circles is within the other, the first is said to touch the other internally, or to have internal contact with it.
7. A segment of a circle is the figure bounded by a chord and one of the two arcs into which the chord divides the circumference.
NOTE. The chord of a segment is sometimes called its base.