C. J.

ON CHORDS.

THEOREM 31. [Euclid III. 3.]

If a straight line drawn from the centre of a circle bisects a chord which does not pass through the centre, it cuts the chord at right angles.

Conversely, if it cuts the chord at right angles, it bisects it.

Let ABC be a circle whose centre is O; and let OD bisect a chord AB which does not pass through the centre.

It is required to prove that OD is perp. to AB.

Conversely. Let OD be perp. to the chord AB

It is required to prove that OD bisects AB.

Proof.

that is,

In the ODA, ODB,

the ▲ ODA, ODB are right angles,

[the

Theor. 7.

Q.E.D.

because the hypotenuse OA = the hypotenuse OB,

and OD is common ;

... DA= DB;

Theor. 18.

COROLLARY 1. The straight line which bisects a chord at right angles passes through the centre.

COROLLARY 2. A straight line cannot meet a circle at more than two points.

For suppose a st. line meets a

circle whose centre is O at the points A and B.

Draw OC perp. to AB.

Then ACCB.

B D

Now if the circle were to cut AB in a third point D, AC would also be equal to CD, which is impossible.

COROLLARY 3. A chord of a circle lies wholly within it.

EXERCISES.

(Numerical and Graphical.)

1. In the figure of Theorem 31, if AB=8 cm., and OD=3 cm., find OB. Draw the figure, and verify your result by measurement.

2. Calculate the length of a chord which stands at a distance 5" from the centre of a circle whose radius is 13".

3. In a circle of 1" radius draw two chords 1.6′′ and 1·2′′ in length. Calculate and measure the distance of each from the centre.

4. Draw a circle whose diameter is 80 cm. and place in it a chord 60 cm. in length. Calculate to the nearest millimetre the distance of the chord from the centre; and verify your result by measurement.

5. Find the distance from the centre of a chord 5 ft. 10 in. in length in a circle whose diameter is 2 yds. 2 in. Verify the result graphically by drawing a figure in which 1 cm. represents 10".

6. AB is a chord 2.4" long in a circle whose centre is O and whose radius is 1.3′′; find the area of the triangle OAB in square inches.

7. Two points P and Q are 3" apart. Draw a circle with radius 1.7" to pass through P and Q. Calculate the distance of its centre from the chord PQ, and verify by measurement.

THEOREM 32.

One circle, and only one, can pass through any three points not in the same straight line.

F

E

Let A, B, C be three points not in the same straight line.

It is required to prove that one circle, and only one, can pass through A, B, and C.

Join AB, BC.

Let AB and BC be bisected at right angles by the lines DF, EG.

Then since AB and BC are not in the same st. line, DF and EG are not par1.

Proof.

Let DF and EG meet in O.

Because DF bisects AB at right angles,

.. every point on DF is equidistant from A and B.

Prob. 14.

Similarly every point on EG is equidistant from B and C.

.. O, the only point common to DF and EG, is equidistant from A, B, and C ;

and there is no other point equidistant from A, B, and C.

.. a circle having its centre at O and radius OA will pass through B and C; and this is the only circle which will pass through the three given points.

Q.E.D.

COROLLARY 1. The size and position of a circle are fully determined if it is known to pass through three given points; for then the position of the centre and length of the radius can be found.

COROLLARY 2. Two circles cannot cut one another in more than two points without coinciding entirely; for if they cut at three points they would have the same centre and radius.

HYPOTHETICAL CONSTRUCTION. From Theorem 32 it appears that we may suppose a circle to be drawn through any three points not in the same straight line.

For example, a circle can be assumed to pass through the vertices of any triangle.

DEFINITION. The circle which passes through the vertices of a triangle is called its circum-circle, and is said to be circumscribed about the triangle. The centre of the circle is called the circum-centre of the triangle, and the radius is called the circum-radius.

EXERCISES ON THEOREMS 31 AND 32.

(Theoretical.)

1. The parts of a straight line intercepted between the circumferences of two concentric circles are equal.

2. Two circles, whose centres are at A and B, intersect at C, D; and M is the middle point of the common chord. Shew that AM and BM are in the same straight line.

Hence prove that the line of centres bisects the common chord at right angles.

3. AB, AC are two equal chords of a circle; shew that the straight line which bisects the angle BAC passes through the centre.

4. Find the locus of the centres of all circles which pass through two given points.

5.

Describe a circle that shall pass through two given points and have its centre in a given straight line.

When is this impossible?

6. Describe a circle of given radius to pass through two given points. When is this impossible?

THEOREM 33. [Euclid III. 9.]

If from a point within a circle more than two equal straight lines can be drawn to the circumference, that point is the centre of the circle.

Let ABC be a circle, and O a point within it from which more than two equal st. lines are drawn to the O, namely OA, OB, OC.

It is required to prove that O is the centre of the circle ABC.

Join AB, BC.

Let D and E be the middle points of AB and BC respectively. Join OD, OE.

.. these angles, being adjacent, are rt. 2.

Theor. 7.

Hence DO bisects the chord AB at right angles, and therefore passes through the centre. Theor. 31, Cor. 1.

Similarly it may be shewn that EO passes through the

centre.

.. O, which is the only point common to DO and EO, must be the centre.

Q.E.D.