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In any parallelogram, the two diagonals bisect each other; and the sum of their squares is equal to the sum of the squares of all the four sides of the parallelogram.
Let ABCD be a parallelogram, of which the diameters are AC and BD; the sum of the squares of AC and BD is equal to the sum of the squares of AB, BC, CD, DA.
Let AC and BD intersect one another in E: and because the vertical angles AED, CEB are equal (8. 1.), and also the alternate angles EAD, ECB (21. 1.), the triangles ADE, CEB have two angles in the one equal to two angles in the other, each to each; but the sides AD and BC, which are opposite to equal angles in these triangles, are also equal (28. 1.); therefore the other sides which are opposite to the equal angles are also equal (26. 1.), viz. AE to EC, and ED to EB. Since, therefore, BD is bisected in E, AB2+AD2-(A. 2.) 2BE2+ 2AE2; and for the same reason, CD2+BC2=2BE2+2EC2=2BE2+ 2AE2, because EC=AE. Therefore AB2+AD2+DC2+BC2=4BE2+ 4AE2. But 4BE2=BD2, and 4AE2=AC2 (2 Cor. 8. 2.) because BD and AC are both bisected in E; therefore AB2+AC2+CD2+BC2=BD2+ AC2.
In the case of the rhombus, the sides AB, BC, being equal, the triangles BEC, DEC, have all the sides of the one equal to the corresponding sides of the other, and are therefore equal: whence it follows that the angles BEC, DEC, are equal; and, therefore, that the two diagonals of a rhombus cut each other at right angles.
1. The radius of a circle is the straight line drawn from the centre to the circumference.
2. A straight line is said to touch a circle, when it meets the circle, and being produced does not cut it.
And that line which has but. one point in common with the circumference, is called a tangent, and the point in common, the point of contact.
3. Circles are said to touch one another, which meet, but do not cut one another.
4. Straight lines are said to be equally distant from the centre of a circle, when the perpendiculars drawn to them from the centre are equal.
5. And the straight line on which the greater pe pendicular falls, is said to be farther from the centre.
6. Any portion of the circumference is called an arc.
The chord or subtense of an arc is the straight line which joins its two extremities.
7. A straight line is said to be inscribed in a circle, when the extremities of it are in the circumference of the circle. And any straight line which meets the circle in two points, is called a secant.
8. A segment of a circle is the figure contained by a straight line, and the arc which it cuts off.
9. An angle in a segment is the angle contained by two straight lines drawn from any point in the circumference of the segment, to the extremities of the straight line which is the base of the segment.
An inscribed triangle, is one which has its three angular points in the circumference.
And, generally, an inscribed figure is one, of which all the angles are in the circumference. The circle is said to circumscribe such a figure.
10. And an angle is said to insist or stand upon the arc intercepted between the straight lines which contain the angle.
This is usually called an angle at the centre. The
angles at the circumference and centre, are
11. The sector of a circle is the figure contained
12. Similar segments of a circle, are those in which the angles are
equal, or which contain equal angles.
PROP. I. THEOR.
A diameter divides a circle and its circumference into two equal parts; and, conversely, the line which divides the circle into two unequal parts is a diameter.
Let AB be a diameter of the circle AEBD, then AEB, ADB are equal in surface and boundary.
Now, if the figure AEB be applied to the figure ADB, their common base AB retaining its position, the curve line AEB must fall on the curve line ADB; otherwise there would, in the one or the other, be points unequally distant from the centre, which is contrary to the definition of a circle.
Conversely. The line dividing the circle into two equal parts is a diameter.
For, let AB divide the circle into two equal parts; then, if the centre is not in AB, let AF be drawn through it, which is therefore a diameter, and consequently divides the circle into two equal parts; hence the portion AEF is equal to the portion AEFB, which is absurd.
COR. The arc of a circle whose chord is a diameter, is a semi-circumference, and the included segment is a semi-circle.
PROP. II. THEOR.
If any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle.
Let ABC be a circle, and A, B any two points in the circumference; the straight line drawn from A to B shall fall within the circle.
Take any point in AB as E; find D the centre of the circle ABC; join AD, DB and DE, and let DE meet the circumference in F. Then, because DA is equal to DB, the angle DAB is equal (3. 1.) to the angle DBA; and because AE, a side of the triangle DAE, is produced to B, the angle DEB is greater (9. 1.) than the angle DAE; but DAE is equal to the angle DBE; therefore the angle DEB is greater than the angle DBE: Now to the greater an
gle the greater side is opposite (12. 1.); DB is therefore greater than DE : but BD is equal to DF; wherefore DF is greater than DE, and the point E is therefore within the circle. The same may be demonstrated of any other point between A and B, therefore AB is within the circle.
COR. Every point, moreover, in the production of AB, is farther from the centre than the circumference.
PROP. III. THEOR.
If a straight line drawn through the centre of a circle bisect a straight line in the circle, which does not pass through the centre, it will cut that line at right angles; and if it cut it at right ungles, it will bisect it.
Let ABC be a circle, and let CD, a straight line drawn through the centre, bisect any straight line AB, which does not pass through the centre, in the point F; It cuts it also at right angles.
Take E the centre of the circle, and join EA, EB. Then because AF is equal to FB, and FE common to the two triangles AFE, BFE, there are two sides in the one equal to two sides in the other: but the base EA is equal to the base EB; therefore the angle AFE is equal (5. 1.) to the angle BFE. And when a straight line standing upon another makes the adjacent angles equal to one another, each of them is a right (7. Def. 1.) angle: Therefore each of the angles AFE, BFE is a right angle; wherefore the straight line CD, drawn through the centre bisecting
AB, which does not pass through the centre, cuts AB at right angles.
Again, let CD cut AB at right angles; CD also bisects AB, that is, AF is equal to FB.
The same construction being made, because the radii EA, EB are equal to one another, the angle EAF is equal (3. 1.) to the angle EBF; and the right angle AFE is equal to the right angle BFE: Therefore, in the two triangles EAF, EBF, there are two angles in one equal to two angles in the other; now the side EF, which is opposite to one of the equal angles in each, is common to both; therefore the other sides are equal (26. 1.): AF therefore is equal to FB.
COR. 1. Hence, the perpendicular through the middle of a chord, passes through the centre; for this perpendicular is the same as the one let fall from the centre on the same chord, since both of them passes through the middle of the chord.
COR. 2. It likewise follows, that the perpendicular drawn through the middle of a chord, and terminated both ways by the circumference of the circle, is a diameter, and the middle point of that diameter is therefore the centre of the circle.
PROP. IV. THEOR.
If in a circle two straight lines cut one another, which do not both pass through the centre, they do not bisect each other.
Let ABCD be a circle, and AC, BD two straight lines in it, which cut one another in the point E, and do not both pass through the centre: AC, BD do not bisect one another.
For if it is possible, let AE be equal to EC, and BE to ED; If one of the lines pass through the centre, it is plain that it cannot be bisected by the other, which does not pass through the centre. But if neither of them pass through the centre, take F the centre of the circle, and join EF and because FE, a straight line through the centre, bisects another AC, which does not pass through the centre, it must cut it at right (3. 3.) angles; wherefore FEA is a right angle. Again, because the straight line FE bisects the straight line BD, which does not pass through the centre, it must cut it at right (3. 3.) angles; wherefore FEB is a right angle: and FEA was shown to be a right angle: therefore FEA is equal to the angle FEB, the less to the greater, which is impossible; therefore AC, BD do not bisect one another.
PROP. V. THEOR.
If two circles cut one another, they cannot have the same centre.
Let the two circles ABC, CDG cut one another in the point B, C ; they have not the same centre.