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(298) To cut a line externally so that the rectangle under the segments shall be equal to a given magnitude.
Let the given magnitude be equal to the square of A, and find a line D whose square is equal to the sum of the squares of A and half the given line BC. From the middle point E of B C take E F equal to this line, and F is the point of external section sought. This is evident from (VI).
Since E F may be taken from the middle point towards either extremity, there are two points of section which solve the problem, equally distant from the middle point. (299) If a line AC be drawn from the vertex A of a triangle to the middle point C of the opposite side, the sum of the squares of the other sides BA and AD is equal to twice the sum of the squares of the bisector A C and half B C of the bisected side.
If A B = A D then ACB is a right angle, and the proposition is evident by XLVII, Book I.
If not, draw the perpendicular A F.
By (XII), the square of A B exceeds the sum of the squares of A C and C B by twice B C CF, or twice DCX CF.
By (XIII), the sum of the squares of A C and C D, or A C and CB, exceeds the square of A D by twice C D X CF.
Hence it appears, that the sum of the squares of the bisector A C and half the base is an arithmetical mean between the squares of the sides A C, A D; and therefore (240) the sum of the squares of the sides is equal to twice the sum of the squares of the bisector and half the bisected side, (300) The sum of the squares of the sides of a quadrilateral figure A B C D, is equal to the sum of the squares of the diagonals together with four times the square of the line E F joining their points of bisection.
Draw B F and D F. The sum of the squares of A B and B C is equal to twice that of B F and CF, and the sum of the squares of A D and D C is equal to twice that of DF and C F (299). But also the sum of the squares of B F and D F is equal to twice that of E F and D E. Hence the proposition is manifest. (301) The sum of the squares of the sides of a parallelogram is equal to that of the diagonals.
For in that case the line E F vanishes, since the diagonals bisect each other (155). (302) If the sum of the squares of the sides of a quadrilateral figure be equal to the sum of the squares of the diagonals, the quadrilateral will be a parallelogram.
For otherwise it would be greater by four times the square of the line EF. (303) If lines be drawn from the three angles of a triangle to the middle points of the opposite sides, three times the sum of the squares of the sides is equal to four times that of the bisectors.
Let A, B, C, be the sides and a, b, c the corresponding bisectors. The sum of the squares of B and C is equal twice the sum of the squares of a and half of A, or twice the sum of the squares of B and C is equal to four times the square of a together with the square of A. In like manner twice the sum of the squares of A and B is equal to four times the square of c together with the square of C, and twice the sum of the squares of A and C is equal to four times the square of 6 together with the square of B. Hence, by adding these equals, and taking the sum of the squares of the sides from both, the proposition follows. (304) If with the middle point C of a finite right line. A B as centre a circle be described, the sums of the squares of the distances of all points in this circle from the extremities of the right line are the same, and equal to twice the sum of the squares of the radius and half the given line.
For the triangles A P B have a common base A B, and the bisectors CP of the base are equal, being radii of the circle. Hence the proposition follows from (299). (305) Hence, if the base of a triangle and the sum of the squares of the sides be given, the locus of the vertex is a circle whose centre is the middle point of the base, and the square of whose radius is half the difference between the square of the base and the sum of the squares of the sides. (306) If a point be assumed within or without a rectangle, the sum of the squares of lines drawn from it to two opposite angles is equal to the sum of the squares of the lines drawn to the other two opposite angles.
This is evident from (299), by considering that the diagonals are equal and bisect each other. (307) In a right angled triangle A B C if a perpendicular BD be drawn, the rectangle A B x DC = the rectangle B D x BC. This might be easily derived from the third book, and still more simply from the sixth book. We shall in the present instance. however, prove it by the 12th proposition of the second book.
Produce A B and D B so that B E=D C, and BF=BC, and draw F E. The triangle BFE is equal in every respect to BCD: E is a right angle. Draw AF. Since E and D are right angles, the rectangle A B x B E - F B x B D (288 Obs.). But FB = B C and B E=DC, : A B x DC=BC X BD, (308) We shall now solve the tenth case of the class of problems mentioned in (273).
Given the difference of the squares of two lines and the rectangle under them to find the lines.
Let a line D C be found (XIV), whose square is equal to the given difference of squares, and on it let a rectangle CE be constructed equal to the given rectangle (XLV, Book I.) Produce C D to A,
so that CA X A D shall be equal to the square of D E
For since the square of A B is equal to CA X AD, the angle A B C is right, :: AB x D C =BD x BC. But AB = DE, :: the rectangle CE=BD X BC, and C E is equal to the given rectangle. It is evident that the difference of the squares of B D and B C is equal to the square of D C, which is equal to the given difference of squares.
I. Equal circles are those whose diameters are equal.
II. A right line is said to touch a
circle when it meets the circle
(311) III. Circles are said to touch one ano
ther which meet but do not cut
(312) IV. Right lines are said to be equally distant from the
centre of a circle when the perpendiculars drawn
to them from the centre are equal, (313) V. And the right line on which the greater perpendicu
lar falls is said to be farther from the centre. (314) VI. A segment of a circle is the figure con
tained by a right line and the part of
(315) VII. An angle in a segment is the angle contained by
two right lines drawn from any point
which is the base of the segment. (316) VIII. An angle is said to stand on the part of the cir
cumference, or the arch, intercepted between
the right lines that contain the angle. (317) IX. A sector of a circle is the figure contained
by two radii and the arch between
them. (318) X. Similar segments of circles are those which contain (319) The subject of the third book of the elements is the properties of the circle, those of the triangle and rectangle having been discussed in the first and second books respectively. (320) The first definition is more properly a theorem. For ' equal circles,' like other equal figures, are those which may be laid one upon the other so as perfectly to coincide. If two circles have equal radii, and the centre of one be laid on the centre of the other, the circles being placed in the same plane, their entire circumferences must be coincident; for if not, a line might be drawn from the common centre to the circumference of one, intersecting that of the other, and thus the circles would have unequal radii, contrary to hyp. (321) In the second definition the meaning of a right line ' cutting a circle' is not explained, and yet it seems as necessary to be defined as é touching a circle. If a right line meet the circumference of a circle, and being produced indefinitely in both directions lie entirely without the circle, it is said to touch it. The line in this case evidently lies entirely on the convex side of the circle.
equal angles. Circles which have the same centre are called concentric circles.
On the other hand a right line which, when produced, meets a circle in two points, is said to cut the circle. The nature of contact and section will appear more plainly as the student proceeds with the third book. (322) The same defect is observable in the third definition. Two circles are said to touch internally when every point of the one, except those at which they meet, is included within the other; and they touch externally when every point of each, except those at which they meet, lie without the other. It will appear by the thirteenth proposition, that contingent circles can only meet at one point. (323) Any part of the circumference of a circle is called an arc of the circle, and the right line which joins its extremities is called its chord. It is evident that two arcs, which together make up the whole circumference, have the same chord.
A diameter is the chord of a semicircle. (324) The distance of a right line from a point is estimated by the perpendicular from the point on the right line. Chords, therefore, are said to be equally or unequally distant, according as the perpendiculars on them from the centre are equal or unequal.
The figure included by an arc and its chord is a segment, and the figure included by an arc and the radii through its extremities is called a sector. (325) It will be proved in Prop. XXI, that all angles inscribed in the same segment of a circle are equal ; and also it will appear, that different segments of the same circle contain unequal angles. Thus a segment becomes as it were characterised by the angle it contains, and those segments of different circles which contain the same angles are said to be similar. In the sixth book we shall show, that such segments bear the same proportion to the entire circles, of which they are parts. (326) A quadrant is a sector whose radii form a right angle. (327) Sectors which have equal radii and equal angles are equal, for they evidently admit of superposition. (328) A sector whose angle is right, is therefore a fourth part of the circle, and its arc is called a ' quadrant.'