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For join E with F, the intersection of the diagonals AC, BD. Because it follows readily from Prop.27. Book I, that these diagonals are equal, and bisect each other, the lines AF, BF, CF, and Bro 21C DF are all equal. Wherefore the \s-2 squares of AE, EC are equivalent .i./*s-, S --to twice the square of AF, and twice the square of EF (II. 22.); and the squares of BE, ED are likewise equivalent to twice the square of BF and twice the same square of EF; consequently, the squares of AF and BF being equal, the squares of AE, EC, are together equivalent to the squares of BE, ED.
PROP. III. THEOR.
If straight lines be drawn from the angular points of a triangle to bisect the opposite sides, thrice the squares of these sides are together equivalent to four times the squares of the bisecting lines.
Let the sides of the triangle ABC be bisected in D, E, and F, and straight lines drawn from these points to the opposite vertices; thrice the squares of the sides AB, BC, and AC are together equivalent to four times the squares of BD, CE and A.F.
For, by Proposition II. 22. the squares of AB, BC are equivalent to twice the square of BD and twice the square of AD, that is, half the square of AC ; the squares of BC, AC are equivalent to twice the squares of CE and half the square of AB; and the squares of AC, AB are equivalent to twice the square of AF and half the square of BC. Whence the squares of the sides of the triangle, repeated twice, are equivalent to twice the squares of BD, CE, and AF, with half the squares of the sides of the triangle. Consequently four times the squares of AB, BC, and AC are equivalent to four times the squares of BD, CE, and AF, with once the squares of AB, BC, and AC; wherefore thrice the squares of the sides AB, BC, and AC are together equivalent to four times the squares of the bisecting lines BD, CE, and AF.
PROP. IV. THEOR.
The squares of the sides of a quadrilateral figure are together equivalent to the squares of its diagonals, together with four times the square of the straight line joining their middle points.
Let ABCD be a quadrilateral figure, in which the straight lines AC, BD, drawn to the opposite corners, are bisected at the points E, F: the squares of AB, BC, CD, and DE, are together equivalent to the squares of AC, BD, together with four times the square of EF. For join EF. And because AC is bisected in F, the squares of AB and BC are equivalent to twice the square of AF and twice the square of BF (II. 22.); and, for the same reason, the squares A --~~2 of CD and DA are equivalent to S.Z.| twice the square of AF and twice To 2^ ro the square of DF. Consequently Z---~! is the squares of all the sides AB, BC, - o so CD, and DA, are equivalent to four 5–3.
times the square of AF—or the square of AC–with twice the squares of BF and of DF. But twice these squares of BF and DF is equivalent (II. 22.) to four times the square of BE, or the square of BD, with four times the square of EF; whence the squares of all the sides of the quadrilateral figure are together equivalent to the squares of its diagonals AC, BD, with four times the square of the straight line EF which joins their points of equal section.
This general theorem seems to have been first given by the illustrious Leonard Euler in the Petersburg Memoirs. It evidently comprehends the twenty-fourth Proposition of this Book ; for when the quadrilateral figure becomes a rhomboid, the diagonals bisect each other, and the middle points E and F coincide; whence the squares of all the sides are equivalent simply to the squares of those diagonals.-If this rhomboid again becomes a rectangle, it will have equal diagonals, and consequently, as in the 10th Proposition of the Second Book, the squares of the sides of a right-angled triangle are equivalent to the square of the hypotenuse.
1. Proposition fifteenth. Hence angles are sometimes measured by a circular instrument, from a point in the circumference, as well as from the centre.
2. Proposition eighteenth. On this proposition depends the construction of amphitheatres; for the visual magnitude of an object is measured by the angle which it subtends at the eye, and consequently the whole extent of the stage, the intermediate objects being purposely darkened or obscured, will be seen with equal advantage by every spectator seated in the ‘same arc of a circle.
3. Proposition twenty-second. To erect a perpendicular, any point D is taken, as in Prop. 34. Book I., and from it a circle is described passing / through C and B; the diameter CDF, by ND its intersection at the point B, determines / N A C B
the position of the perpendicular BF. To let fall a perpendicular, draw to AB any straight line FC, which bisect in D, and from this point as a
centre describe a circle through the points C, B and F; FB is the perpendicular required.
4. To this Book may be subjoined some useful propositions.
PROP. I. THEOR.
The inclination of two straight lines is equal to the angle terminated at the circumference by the sum or difference of the arcs which they intercept, according as their vertex is within or without the circle.
If the two straight lines AB and CD intersect each other in the point E within a circle; the angle AED which they form, is equal to an angle at the circumference and standing on the sum of the intercepted arcs AD and BC. For draw the chord BF parallel to CD. Because ED and BF are parallel, the angle AED (I.22.) 2. is equal to the interior angle ABF, o which stands on the arc AF; but since / E. the chords BF and CD are parallel, the N arc BC is equal to DF (III. 18.) and \ y consequently the arc AF, which termi- \ 'D nates at the circumference an angle equal to AED, is the sum of the two intercepted arcs AD and BC. Again, if the straight lines AB and CD meet at E, without the circle, their inclination AED is equal to an angle at the circumference, having for its base the excess of the arc AD above BC. For BF being drawn parallel to CD, the arc BC is equal to FD, and consequently the arc AF is the excess of AD above BC; but the angle ABF which stands on AF, is equal to the interior angle AED.
Cor. Hence if two chords intersect each other at right angles within a circle, the opposite intercepted arcs are equal to the semicircumference.
This proposition is of some utility in practice, for an angle may be hence measured by help of a circular protractor, without the trouble of applying the centre to its vertex or the point of concourse of the sides. The same principle is likewise applicable to the construction of some optical instruments, adapted to measure lateral angles by the intersection of micrometer wires.
PROP. II. THEOR.
If a circle be described on the radius of another circle, any straight line drawn from the point where they meet to the outer
circumference, is bisected by the interior one. l
Let AEC be a circle described on the radius AC of the circle ADB, and AD a straight line
drawn from A to terminate in the ex- T) terior circumference; the part AE in the smaller circle is equal to the part #3
ED intercepted between the two cir- A NL/
a semicircle, the angle contained in it is a right angle (III. 19.); consequently the straight line CE, drawn from the centre C, is perpendicular to the chord AD, and therefore (III. 4.) bisects it.
PROP. III. THEOR.
If, on each side of any point in the circumference of a circle, equal arcs be repeated; the chords which join the opposite points of section will be together equal to the last chord extended till it meets a straight line drawn through the middle point and either extremity of the first chord.