PROPOSITION X. THEOREM.

Two circles cannot have more than two points common to both, without coinciding entirely.

If it be possible, let ABC and ADE be two Os which have more than two pts. in common, as A, B, C.

Join AB, BC.

Then AB is a chord of each circle,

..the centre of each circle lies in the straight line, which bisects AB at right angles ;

and BC is a chord of each circle,

III. 1.

.. the centre of each circle lies in the straight line, which bisects BC at right angles.

III. 1.

.. the centre of each circle is the point, in which the two straight lines, which bisect AB and BC at right angles, meet. .. the os ABC, ADE have a common centre, which is impossible; III. 5 and 6.

.. two Os cannot have more than two pts. common to both.

Q. E. D.

NOTE. We here insert two Propositions, Eucl. III. 25 and IV. 5, which are closely connected with Theorems I. and x. of this book. The learner should compare with this portion of the subject the note on Loci, p. 103.

PROPOSITION A. PROBLEM. (Eucl. III. 25.)

An arc of a circle being given, to complete the circle of which it is a part.

D

B

Let ABC be the given arc.

It is required to complete the of which ABC is a part.
Take B, any pt. in arc ABC, and join AB, BC.
From D and E, the middle pts. of AB and BC,
draw DO, EO, 1s to AB, BC, meeting in O.
Then AB is to be a chord of the O,

.. centre of the C lies in DO;
and BC is to be a chord of the O,

III. 1.

.. centre of the

lies in EO.

III. 1.

Hence O is the centre of the

of which ABC is an arc,

and if a be described, with centre O and radius OA, this

will be the

required.

Q. E. F.

PROPOSITION B. PROBLEM. (Eucl. iv. 5.)
To describe a circle about a given triangle.

B

D

Let ABC be the given ▲.

It is required to describe a about the ▲.

From D and E, the middle pts. of AB and AC, draw DO, EO, 1s to AB, AC, and let them meet in O.

Hence O is the centre of the which can be described

about the ▲, and if a

OA, this will be the

be described with centre O and radius required.

Q. E. F.

Ex. 1. If BAC be a right angle, shew that O will coincide with the middle point of BC.

Ex. 2. If BAC be an obtuse angle, shew that O will fall on the side of BC remote from A.

PROPOSITION XI. THEOREM.

If one circle touch another internally at any point, the centre of the interior circle must lie in that radius of the other circle which passes through that point of contact.

E

Let the ADE touch the ABC internally, and let A be a pt. of contact.

Find the centre of O ABC, and join OA.

Then must the centre of ADE lie in the radius OA.

For if not, let P be the centre of ADE.

Join OP, and produce it to meet the Oces in D and B. Then P is the centre of ADE, and from O are drawn to the Oce of ADE the st. lines OA, OD, of which OD passes through P,

.. OD is greater than OA.
But OA=0B;

which is impossible.

.. OD is greater than OB,

III. 8, Cor.

.. the centre of O ADE is not out of the radius OA.

.. it lies in OA.

Q. E. D.

PROPOSITION XII. THEOREM.

If two circles touch one another externally at any point, the straight line joining the centre of one with that point of contact must when produced pass through the centre of the other.

BD

ӨӨ

Let ABC touch O ADE externally at the pt. A.
Let O be the centre of O ABC.

Join OA, and produce it to E.

Then must the centre of O ADE lie in AE.

For if not, let P be the centre of ○ ADE.

Join OP meeting the Os in B, D; and join AP. Then OB=0A,

and PD=AP,

.. OB and PD together=0A and AP together;
.. OP is not less than OA and AP together.
But OP is less than OA and AP together,

which is impossible;

.. the centre of O ADE cannot lie out of AE.

Ex. Three circles

centres are A, B, C.

I. 20.

Q. E. D.

touch one another externally, whose Shew that the difference between AB

and AC is half as great as the difference between the diameters

of the circles, whose centres are B and C.