3. AB, CD are chords of a circle which cut at a constant angle. Prove that the sum of the arcs AC, BD remains constant, whatever may be the position of the chords.

4. If the diameter of a circle be one of the equal sides of an isosceles triangle, prove that its circumference will bisect the base of the triangle.

5. Circles are described on two sides of a triangle as diameters. Prove that they will intersect on the third side

or third side produced.

6. Any number of chords of a circle are drawn through a point on its circumference: find the locus of their middle points.

7. If through any point, within or without a circle, lines are drawn to cut the circle, prove that the locus of the middle points of the chords so formed is a circle.

8. In any inscribed hexagon the sum of any three alternate angles is equal to four right angles.

SECTION IV. A.

TANGENTS (treated directly).

Def. 10. A secant is a straight line of unlimited length which meets the circumference of a circle in two points.

THEOREM 18.

Every straight line through a point on the circumference of a circle meets it in one other point, except the straight line perpendicular to the radius at the point *.

Part. En. Let A be a circle, B its centre, and BC a radius; and let CD be a line through C perpendicular to the radius BC, and CE any other line;

it is required to prove that CE meets the circle in one point other than C, and that CD does not.

Proof. Because BC is perpendicular to CD,

therefore BC is the shortest line from B to the line CD: (1. 19.)

therefore every point in CD other than C is at a distance from B greater than BC, that is than the radius of the circle.

Therefore no point in CD

other than C is on the circumference.

* Euclid, III. 16.

Again, from B draw BF perpendicular to CE, and BG making an angle with BF, on the side remote from C, equal to CBF, and meeting CE in G.

Then because BC and BG are straight lines from B to the straight line CE making equal angles with the perpendicular BF upon it, they are equal; (1. 19.)

that is, BG is equal to the radius of the circle,

and therefore G lies upon the circumference; that is, the line CE meets the circle again in G.

Def. II. A straight line which, though produced indefinitely, meets the circumference of a circle in one point only is said to touch, or to be a tangent to, the circle.

Def. 12. The point at which a tangent meets the circumference is called the point of contact.

The following are immediate consequences of Theorem 18.

(a) One and only one tangent can be drawn to a circle at a given point on the circumference.

(b) The tangent to a circle is perpendicular to the radius drawn to the point of contact.

(c) The centre of a circle lies in the perpendicular to the tangent at the point of contact.

(d) The straight line drawn from the centre perpendicular to the tangent passes through the point of contact.

Obs. On the relative position of a straight line and a circle.

A straight line will cut a circle, touch it, or not meet it at all, according as its distance from the centre is less than, equal to, or greater than the radius.

The several converses of these statements follow by the Rule of Conversion.

THEOREM 19..

Each angle contained by a tangent and a chord drawn from the point of contact is equal to the angle in the alternate segment of the circle*.

Part. En. Let DBC be a tangent to the circle A at the point B, and let BE be a line through B meeting the circle again in E;

it is required to prove that the angles contained by DBC and BE are

D

B

equal to the angles in the alternate segments upon BE.

Proof. Draw BF the diameter through B;

then BF will be at right angles to DC;

(Th. 18.) and join F, E and B to any point G in the minor arc BE.

Then because FGB is an angle in a semicircle it is a right angle;

and therefore the angle FGB is equal to the angle FBD; also the angle EGF is equal to the angle EBF in the same segment;

*Eucl. III. 32.

therefore the whole angle EGB is equal to the whole angle

also the angle FHE is equal to the angle FBE in the same segment;

therefore the remaining angle EHB is equal to the remaining angle EBC.

Obs. Having proved Th. 19 so far as it relates to either of the two angles EBC, EBD, its truth as it relates to the other follows at once from Th. 17, since the angle in the conjugate segment and the remaining angle at B are respectively supplementary to the two equal angles.

THEOREM 20.

Two tangents, and only two, can be drawn to a circle from an external point.

Part En. Let A be a point external to the given circle BCD; it is required to prove that two, and only two, straight lines can be drawn from A to touch the circle BCD.

W.

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