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5. Circles are said to touch mutually, if they meet, but do not cut each other.

6. The point where a straight line touches a circle, or one circle touches another, is called the point of contact.

7. A straight line is said to be inflected from a point, when it terminates in another straight line, or at the circumference of a circle.

PROP. I. THEOR.

A circle is bisected oy its diameter.

The circle ADBE is divided into two equal portions,

by the diameter. AB.

For let the portion ADB be reversed and applied to

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D of the former must meet with a corresponding point of the latter, and consequently the two portions ADB and

AEB will entirely coincide.

Cor. The portion ADB limited by a diameter, is thus a semicircle, and the arc ADB is a semicircumference.

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ly two equal straight lines, such as CD and CE, can be drawn to AB (I. 17. cor.) the circle described from C through the point D will cross AB again only at E.

PROP. III. THEOR.

The chord of an arc lies wholly within the circle.

The straight line AB which joins two points A, B in t circumference of a circle, lies wholly within the figure. For, from the centre C, draw CD to any point in AB, and join CA and CB. Because CDA is the exterior angle of the triangle CDB, it is greater (I. 8.) than the interior CBD or CBA; but CBA, being (I. 10.) equal to CAB or CAD, the angle CDA is consequently greater than CAD, and its opposite side CA (I, 13.) greater than CD, or CD is less than CA, and therefore the point D must lie within the circle. Cor. Hence a circle is concave towards its centre.

PROP. IV. THEOR.

A straight line drawn from the centre of a circle at right angles to a chord, likewise bisects it; and, conversely, the straight line which joins the centre with the middle of a chord, is perpendicular to it.

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The perpendicular let fall from the centre C upon the chord AB, cuts it into two equal parts AD, DB. For join CA, CB: And, in the triangles ACD, BCD, the side AC is equal to CB, CD is common to both, and the right angle ADC is equal to BDC; these triangles having thus their corresponding angles at A and B both acute, are equal ZN (I. 21.) and consequently the side AD A is equal to BD. Again, let AD be equal to BD ; the bisecting line CD is at right angles to AB. For join CA, CB. The triangles ACD and BCD, having the sides AC, AD equal to CB, BD, and the remaining side CD common to both, are equal (I. 2.), and consequently the angle CDA is equal to CDB, and each of them a right angle. Cor. Hence a straight line cutting two concentric circles has equal portions intercepted by their circumferences.

PROP. V. THEOR.

A straight line which bisects a chord at right angles, passes through the centre of the circle.

If the perpendicular FE bisect a chord AB, it will pass through G

the centre of the circle.
For in FE take any point D, and
join DA and DB. The triangles

ADC and BDC, having the side *NSD7*.

AC equal to BC, CD common, and IF

the right angle ACD equal to BCD, are equal (I. 3.), and consequently the base AD is equal to BD. The point D is, therefore, the centre of a circle described through A and B; and thus the centres of the circles that can pass through A and B are all found in the straight line EF. The centre G of the circle AEBF must hence occur in that perpendicular. Cor. The centre of a circle may hence be found by bisecting the chord AB by the diameter EF (I. 7.), and bisecting this again in G.

PROP. VI. THEOR.

The diameter is the greatest line that can be inflected within a circle.

The diameter AB is greater than any chord DE.

For join CD and CE. The two sides DC and EC of the triangle DCE are together greater than the third side DE (I. 14.); but DC and CE are equal to AC and CB, or to the whole diameter AB. Wherefore AB is greater than DE.

PROP. VII. THEOR.

If from any eccentric point, two straight lines be drawn to the circumference of a circle ; the one which passes nearer the centre, is greater than that which lies more remote.

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