Now sum of sqq. on OP, PC=sum of sqq. on OQ, QE.

But sq. on PC is greater than sq. on QE;

.. sq. on OP is less than sq. on OQ;

.. OP is less than OQ;

and .. CD is nearer to the centre than EF. Q. E. D.

Ex. 1. Draw a chord of given length in a given circle, which shall be bisected by a given chord.

Ex. 2. If two isosceles triangles be of equal altitude, and the sides of one be equal to the sides of the other, shew that their bases must be equal.

Ex. 3. Any two chords of a circle, which cut a diameter in the same point and at equal angles, are equal to one another.

DEF. IX. A straight line is said to be a TANGENT to, or to touch, a circle, when it meets and, being produced, does not cut the circle.

From this definition it follows that the tangent meets the circle in one point only, for if it met the circle in two points it would cut the circle, since the line joining two points in the circumference is, being produced, a secant. (III. 2).

DEF. X. If from any point in a circle a line be drawn at right angles to the tangent at that point, the line is called a NORMAL to the circle at that point.

DEF. XI. A rectilinear figure is said to be described about a circle when each side of the figure touches the circle; and the circle is said to be inscribed in the figure.

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PROPOSITION XVI. THEOREM.

The straight line drawn at right angles to the diameter of a circle, from the extremity of it, is a tangent to the circle.

Let ABC be a O, of which the centre is 0, and the diameter AOB.

In the same way it may be shewn that every point in DE, or DE produced in either direction, except the point B, lies outside the O;

.. DE is a tangent to the O.

Def. IX.

Q. E. D,

PROPOSITION XVII. PROBLEM.

To draw a straight line from a given point, either WITHOUT or on the circumference, which shall touch a given circle.

Let A be the given pt., without the © BCD.

Let O be the centre of © BCD, and join OA. Bisect OA in E, and with centre E and radius EO, describe a ABOD, cutting the given in B and D. Join AB, AD. These are tangents to the ○ BCD.

Join BO, BE.

Then OE=BE, :. ▲ OBE= ↳ BOE;

.. LAEB=twice OBE;

and ·: AE=BE, :. ▲ ABE= L BAE;

.. LOEB twice ABE;

=

L. 32.

I. 32.

.. sum of 48 AEB, OEB = twice sum of 48 OBE, ABE, that is, two right angles=twice ▲ OBA;

.. LOBA is a right angle,

I. 16.

and .. AB is a tangent to the ○ BCD. Similarly it may be shewn that AD is a tangent to O

BCD.

Next, let the given pt. be on the ○ce of the ©, as B.
Then, if BA be drawn to the radius OB,
BA is a tangent to the C at B.

I. 16.

Q. E. D.

Ex. 1. Shew that the two tangents drawn from a point without the circumference to a circle are equal.

Ex. 2. If a quadrilateral ABCD be described about a circle, shew that the sum of AB and CD is equal to the sum of AC and BD.

PROPOSITION XVIII. THEOREM.

If a straight line touch a circle, the straight line drawn from the centre to the point of contact must be perpendicular to the line touching the circle.

B

F

Let the st. line DE touch the

ABC in the pt. C.

Find the centre, and join OC.

Then must OC be 1 to DE.

For if it be not, draw OBF to DE, meeting the Oce in B. Then OFC is a rt. angle,

.. OCF is less than a rt. angle,

I. 17.

and.. OC is greater than OF.

I. 19.

But OC=OB,

:. OB is greater than OF, which is impossible;

..OF is not to DE, and in the same way it may be shewn that no other line drawn from O, but OC, is 1 to DE;

Ex. If two straight lines intersect, the centres of all circles touched by both lines lie in two lines at right angles to each other.

NOTE. Prop. XVIII. might be stated thus:-All radii of a circle are normals to the circle at the points where they meet the circumference.

PROPOSITION XIX. THEOREM.

If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the touching line, the centre of the circle must be in that line.

Let the st. line DE touch the ABC at the pt. C, and from Clet CA be drawn to DE.

Then

Then must the centre of the be in CA.

For if not, let F be the centre, and join FC.

DCE touches the O, and FC is drawn from centre

to pt. of contact,

.. FCE is a rt. angle.

But ACE is a rt. angle;

:. ▲ FCE= L ACE, which is impossible.

III. 18.

In the same way it may be shewn that no pt. out of CA can be the centre of the ;

.. the centre of the lies in CA.

Q. E. D.

Ex. Two concentric circles being described, if a chord of the greater touch the less, the parts intercepted between the two circles are equal.

NOTE. Prop. XIX. might be stated thus:-Every normal to a cir cle passes through the centre.