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

B

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, is to AB, AC, and let them meet in O.

Hence O is the centre of the O which can be described about the ▲, and if a C be described with centre O and radius OA, this will be the 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.

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.

P

Let the O 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;

III. 8, Cor.

.. OD is greater than OB,

which is impossible.

.. 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.

G

BD

A

Э

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

Join OA, and produce it to E.

Then must the centre of ○ ADE lie in AE.

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

付

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.

I. 20.

Q. E. D.

Ex. Three circles

centres are A, B, C.

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.

PROPOSITION XIII. THEOREM.

One circle cannot touch another at more points than one, whether it touch it internally or externally.

First let the ADE touch the ○ ABC internally at pt. A.

Then there can be no other point of contact.

B

Take O the centre of O ABC

Then P, the centre of o ADE, lies in OA.

III. 11.

Take any pt. E in the Oce of the O ADE, and join OE.

Then from O, a pt. within or without the © ADE, two lines OA, OE are drawn to the Oce, of which OA passes through the centre P,

.. OA is greater than OE,

and.. E is a point within the ABC.

III. 8, Cor.

Post.

Similarly it may be shewn that every pt. of the Oce of the

O ADE, except A, lies within the ABC;

.. A is the only point at which the Os meet.

Next, let the Os ABC, ADE touch externally at the pt. A.

Then there can be no other point of contact.

Take O the centre of the O ABC.

Then P, the centre of the O ADE, lies in OA produced.

III. 12.

Take any pt. D in the Oce of the O ADE, and join OD. Then from O, a pt. without the ○ ADE, two lines OA, OD are drawn to the Oce, of which OA when produced passes through the centre P,

.. A is the only point at which the Os meet.

Q. E. D.

DEF. VIII. The DISTANCE of a chord from the centre is

measured by the length of the perpendicular drawn from the centre to the chord.