Conversely:

Of any number of chords in a circle the greater is nearer to the centre than the less.

In the circle AGB let the chord HK be > FG.
Then shall HK be nearer to the centre C than FG.

The same construction being made,

.. CH CF, ... square on CH = square on CF;

=

.. squares on CX, XH are together=squares on CY, YF; (1.35) but since HK is > FG; .. XH is > YF;

... square on XH is > square on YF;

... square on CX is < square on CY;

.. CX is < CY;

.. HK is nearer to the centre C than FG.

DEFINITION.

An arc of a circle is a part of the circumference.

PROBLEM A.

An arc of a circle being given, to find the centre of the circle of which it is an arc.

K

Take any three points P, Q, R in the given arc.
Join PQ, QR:

bisect PQ in H and QR in K.

Through H, K draw straight lines to PQ, QR respectively.

Then the centre of the must lie in each of these straight lines;

and must therefore be their point of intersection.

(III. 2)

PROPOSITION VI.

The straight line drawn through any point in the circumference of a circle perpendicular to the radius at that point falls without the circle: also through that point no other straight line can be drawn which does not cut the circle.

T

B

A

Let A be any point in the circumference of the AB.
Join A with the centre C;

and through A draw ST1 to CA.
Then ST shall fall without the .

For, if possible, let ST cut the circle in B.

Join BC.

Then. CB = CA;

..

LCBA = L CAB;

(1.2)

but

CAB is a right ; .. 4 CBA is a right 4;

... two of the 4s of a ▲ are right ≤ s, which is impossible; (1.13) .. SAT does not cut the O.

Similarly it may be shewn that it has no other point besides A in common with the circumference.

Moreover, besides SAT no other straight line can be drawn through A so as not to cut the circle.

I

For, if possible, let YAX be so drawn: let fall CH on YAX cutting the O in Q.

A straight line which meets a circle and which, if produced, will not cut it, is said to touch the circle.

Conversely:

If a straight line touch a circle the radius through the point of contact is perpendicular to the tangent.

T

A

For through the point of contact only one straight line can be drawn touching the circle, and that straight line is to the radius.

Also, If a straight line touch a circle the perpendicular to the tangent drawn from the point of contact shall pass through the centre.

For from the point of contact only one straight line can be drawn to the tangent, and that through the centre is

L to it.