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.

E

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 .

Join AB, BC, and draw OD, OE 1 to AB, BC.

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, Part I. 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.

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 ○ lies in DO; and BC is to be a chord of the

.. centre of the lies in EO.

Hence O is the centre of the

and if a

will be the

III. 1.

III. 1.

of which ABC is an arc,

be described, with centre O and radius OA, this

required.

Q. E. F.

PROPOSITION B. PROBLEM. (Eucl. IV. 5.)

It is required to describe a about the ▲.

From D and E, the middle pts. of AB and BC, draw DO, EO, 18 to AB, BC, and let them meet in O.

Hence O is the centre of the which can be described be described with centre O and

about the ▲, and if a radius OA, this will be the

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.

B

P

E

Let the ADE touch the ○ ABC internally, and let A

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.

III. 8, Cor.

But OA=OB;

.. OD is greater than OB,

which is impossible.

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