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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.

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Let ABC be a O, 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.

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.. 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.

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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=A0;

.. 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 c 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 centre.

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.

Let O be a pt. in the

D

ABC from which more than two st.

lines OA, OB, OC, drawn to the Oce, are cqual.

Then must O be the centre of the .

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;

.. the centre of the O is in DO.

Similarly it may be shown that

the centre of the is EO;

.. O is the centre of the O.

I. E. Cor. p. 43.

III. 1.

Q. E. D.

PROPOSITION X. THEOREM.

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

D

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

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,

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

required.

Q. E. F.

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