*THEOREM 36. [Euclid III. 7.]

If from any internal point, not the centre, straight lines are drawn to the circumference of a circle, then the greatest is that which passes through the centre, and the least is the remaining part of that diameter.

And of any other two such lines the greater is that which subtends the greater angle at the centre.

Let ACDB be a circle, and from P any internal point, which is not the centre, let PA, PB, PC, PD be drawn to the Oce, so that PA passes through the centre O, and PB is the remain ing part of that diameter. Also let the POC at the centre subtended by PC be greater than the ▲ POD subtended by PD.

It is required to prove that of these st. lines

(i) PA is the greatest,

(ii) PB is the least,

(iii) PC is greater than PD.

Join OC, OD.

Proof. (i) In the APOC, the sides PO, OC are together greater than PC.

But OCOA, being radii ;

.. PO, OA are together greater than PC;

that is, PA is greater than PC.

Theor. 11.

Similarly PA may be shewn to be greater than any other st. line drawn from P to the Oce;

.. PA is the greatest of all such lines.

(ii) In the OPD, the sides OP, PD are together greater than OD.

But OD=OB, being radii;

.. OP, PD are together greater than OB.

Take away the common part OP;

then PD is greater than PB.

ce

Similarly any other st. line drawn from P to the Oe may be shewn to be greater than PB;

1. All circles which pass through a fixed point, and have their centres on a given straight line, pass also through a second fixed point.

2. If two circles which intersect are cut by a straight line parallel to the common chord, shew that the parts of it intercepted between the circumferences are equal.

3. If two circles cut one another, any two parallel straight lines drawn through the points of intersection to cut the circles are equal.

4. If two circles cut one another, any two straight lines drawn through a point of section, making equal angles with the common chord, and terminated by the circumferences, are equal.

5. Two circles of diameters 74 and 40 inches respectively have a common chord 2 feet in length: find the distance between their centres. Draw the figure (1 cm. to represent 10") and verify your result by

measurement.

6. Draw two circles of radii 1·0′′ and 1.7", and with their centres 2.1" apart. Find by calculation, and by measurement, the length of the common chord, and its distance from the two centres.

*'THEOREM 37. [Euclid III. 8.]

If from any external point straight lines are drawn to the circumference of a circle, the greatest is that which passes through the centre, and the least is that which when produced passes through the centre.

And of any other two such lines, the greater is that which subtends the greater angle at the centre.

Let ACDB be a circle, and from any external point P let the lines PBA, PC, PD be drawn to the O, so that PBA passes through the centre O, and so that the POC subtended by PC at the centre is greater than the ▲ POD subtended by PD.

It is required to prove that of these st. lines

Proof. (i) In the ▲ POC, the sides PO, OC are together

greater than PC.

But OC=OA, being radii ;

.. PO, OA are together greater than PC;

that is, PA is greater than PC.

Similarly PA may be shewn to be greater than any other st line drawn from P to the Oce;

that is, PA is the greatest of all such lines.

(ii) In the POD, the sides PD, DO are together greater than PO.

But OD OB, being radii;

.. the remainder PD is greater than the remainder PB.

Similarly any other st. line drawn from P to the O may be shewn to be greater than PB;

that is, PB is the least of all such lines.

(iii) In the APOC, POD,

(PO is common,

because OC=OD, being radii ;

but the POC is greater than the

.. PC is greater than PD.

POD;

Theor. 19.

Q.E.D.

EXERCISES.

(Miscellaneous.)

1. Find the greatest and least straight lines which have one extremity on each of two given circles which do not intersect.

2. If from any point on the circumference of a circle straight lines are drawn to the circumference, the greatest is that which passes through the centre; and of any two such lines the greater is that which subtends the greater angle at the centre.

3. Of all straight lines drawn through a point of intersection of two circles, and terminated by the circumferences, the greatest is that which is parallel to the line of centres.

4. Draw on squared paper any two circles which have their centres on the x-axis, and cut at the point (8, -11). Find the coordinates of their other point of intersection.

5. Draw on squared paper two circles with centres at the points (15, 0) and (-6, 0) respectively, and cutting at the point (0, 8). Find the lengths of their radii, and the coordinates of their other point of intersection.

6. Draw an isosceles triangle OAB with an angle of 80° at its vertex O. With centre O and radius OA draw a circle, and on its circumference take any number of points P, Q, R, on the same side of AB as the centre. Measure the angles subtended by the chord AB at the points P, Q, R,.... Repeat the same exercise with any other given angle at O. What inference do you draw?

ON ANGLES IN SEGMENTS, AND ANGLES AT THE CENTRES AND CIRCUMFERENCES OF CIRCLES.

THEOREM 38. [Euclid III. 20.]

The angle at the centre of a circle is double of an angle at the circumference standing on the same arc.

Let ABC be a circle, of which O is the centre; and let BOC be the 'angle at the centre, and BAC an angle at the ○°, standing on the same arc BC.

It is required to prove that the L BOC is twice the L BAC.

Proof.

Join AO, and produce it to D.

In the OAB, because OB = OA,
... the OAB = the ▲ OBA.

.. the sum of the 2o OAB, OBA = twice the ▲ OAB.

But the ext. 4 BOD the sum of the 28 OAB, OBA;

. the

=

BOD = twice the ▲ OAB.

Similarly the DOC = twice the OAC.

.., adding these results in Fig. 1, and taking the difference in Fig. 2, it follows in each case that

the BOC=twice the BAC.

Q.E.D.