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 O, and the diameter AOB.

Through B draw DE at right angles to AOB.
Then must DE be a tangent to the .

I. 11.

In the same way it may be shewn that every point in DE, or DE produced in either direction, except the point B, lies without the O;

.. DE is a tangent to the .

Def. 9.

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.
Take O the centre of BCD, and join OA.

Bisect OA in E, and with centre E and radius EO describe

O ABOD, cutting the given in B and D.

Join AB, AD. These are tangents to the BCD.

.. sum of 4s AEB, OEB=twice sum of 4s OBE, ABE, that is, two right angles=twice ▲ OBA ;

Similarly it may be shewn that AD is a tangent to
Next, let the given pt. be on the Oce of the O, as B.
Then, if BA be drawn to the radius OB,

BA is a tangent to the O at B.

III. 16.
BCD.

III. 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.

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.

A

B

F

Let the st. line DE touch the O ABC in the pt. C.

Find O the centre, and join OC.

Then must OC be 1 to DE.

For if it be not, draw OBF1 to DE, meeting the Oce in B.

.. 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;

.. OC is 1 to DE.

Q. E. D.

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.

11

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

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 of the chord, intercepted between the two circles, are equal.

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

PROPOSITION XX. THEOREM.

The angle at the centre of a circle is double of the angle at the circumference, subtended by the same arc.

Let ABC be a O, O the centre,

BC any arc, A any pt. in the Oce.

Then must / BOC = twice L BAC.

First, suppose O to be in one of the lines containing the L BAC.

..4 OCA = LOAC;

.. sum of 48 OCA, OAC = twice ▲ OAC.

But / BOC sum of 4 s OCA, OAC,

=

.. BOC twice 4 OAC.

that is, BOC- twice ▲ BAC.

I. A.

I. 32.