NOTE 1. On the Contact of Circles.

DEF. VII. Circles are said to touch each other, which meet but do not cut each other.

One circle is said to touch another internally, when one point of the circumference of the former lies on, and no point without, the circumference of the other.

Hence for internal contact one circle must be smaller than the other.

Two circles are said to touch externally, when one point of the circumference of the one lies on, and no point within the circumference of the other.

N.B. No restriction is placed by these definitions on the number of points of contact, and it is not till we reach Prop. XIII. that we prove that there can be but one point of contact.

PROPOSITION VI. THEOREM.

If one circle touch another internally, they cannot have the same centre.

and let A be a point of contact.

Then some point E in the Oce ADE lies within

ABC.

Def. 7.

If it be possible, let O be the common centre of the two Join OA, and draw OEC, meeting the Oces in E and C. is the the centre of O ABC,

Then

.. OA=OC;

and is the centre of ADE,

s.

I. Def. 13.

.. OA=OE.

Hence OE=OC, which is impossible;

I. Def. 13.

.. O is not the common centre of the two Os.

PROPOSITION VII. THEOREM.

If from any point within a circle, which is not the centre, straight lines be drawn to the circumference, the greatest of these lines is that which passes through the centre.

B

Let ABC be a C, of which O is the centre.

From P, any pt. within the O, draw the st. line PA, passing through O and meeting the Oce in A.

Then must PA be greater than any other st. line,
drawn from P to the Oce.

For let PB be any other st. line, drawn from P to meet the Oce in B, and join BO.

.. AP sum of BO and OP.

But the sum of BO and OP is greater than BP,

and.. AP is greater than BP.

I. 20.

Q. E. D.

Ex. 1. If AP be produced to meet the circumference in D, shew that PD is less than any other straight line that can be drawn from P to the circumference.

Ex. 2. Shew that PB continually decreases, as B passes from A to D.

Ex. 3. Shew that two straight lines, but not three, that shall be equal, can be drawn from P to the circumference,

PROPOSITION VIII. THEOREM.

If from any point without a circle straight lines be drawn to the circumference, the least of these lines is that which, when produced, passes through the centre, and the greatest is that which passes through the centre.

B

Let ABC be a ©, of which O is the centre.

From P any pt. outside the O, draw the st. line PAOC, meeting the Oce in A and C.

Then must PA be less, and PC greater, than any other st. line drawn from P to the Oce.

For let PB be any other st. line drawn from P to meet the Oce in B, and join BO.

Then sum of PB and BO is greater than OP,

I. 20.

.. sum of PB and BO is greater than sum of AP and AO.

But BO=AO;

.. PB is greater than AP.

Again PB is less than the sum of PO, OB,

.. PB is less than the sum of PO, OC;
.. PB is less than PC.

I. 20.

Q. E. D.

Ex. 1. Shew that PB continually increases as B passes from A to C.

Ex. 2. Shew that from P two straight lines, but not three, that shall be equal, can be drawn to the circumference.

NOTE. From Props. VII. and VIII. we deduce the following Corollary, which we shall use in the proof of Props. XI. and XIII.

COR. If a point be taken, within or without a circle, of all straight lines drawn from it to the circumference, the greatest is that which meets the circumference after passing through the cent

PROPOSITION IX. THEOREM.

If a point be taken within a circle, from which there fall more than two equal straight lines to the circumference, that point is the centre of the circle.

B

D

Let O be a pt. in the ABC from which more than two st. lines OA, OB, OC, drawn to the Oce, are equal.

Then must O be the centre of the O.

Join AB, BC, and draw OD, OE 1 to AB, BC.
Then OA=OB, and OD is common,

in the right-angled As AOD, BOD,

.. AD=DB;

I. E. Cor. p. 43.

..the centre of the O is in DO.

III. 1.

Similarly it may be shown that

the centre of the O is EO;

.. O is the centre of the O.