PROPOSITION VI. THEOREM.

182. CONVERSELY: In the same circle, or equal circles, equal chords subtend equal arcs.

In the equal circles ABP and A'B' P', let chord RS

..AROS= ▲ R'O'S',

(three sides of the one being equal to three sides of the other).

:.40 = 20',

(being homologous ▲ of equal ▲).

.. arc RS = arc R' S',

Hyp.

176

§ 176

$108

$ 179

(in the same O, or equal, equal at the centre intercept equal arcs on the

PROPOSITION VII. THEOREM.

183. The radius perpendicular to a chord bisects the chord and the arc subtended by it.

Let A B be the chord, and let the radius CS be perpendicular to AB at the point M.

(being radii of the same O);

..AAC B is isosceles,

(the opposite sides being equal) ;

.. LCS bisects the base A B and the Z C,

§ 84

§ 113

(the drawn from the vertex to the base of an isosceles ▲ bisects the base and

(equal at the centre intercept equal arcs on the circumference).

Q. E. D.

184. COROLLARY. The perpendicular erected at the middle of a chord passes through the centre of the circle, and bisects the arc of the chord.

PROPOSITION VIII. THEOREM.

185. In the same circle, or equal circles, equal chords are equally distant from the centre; and of two unequal chords the less is at the greater distance from the centre.

A

m

H

B

F

E

In the circle ABEC let the chord A B equal the chord CF, and the chord CE be less than the chord C F. Let OP, OH, and OK be is drawn to these chords from the centre 0.

..LOK will intersect CF in some point, as m.

(a is the shortest distance from a point to a straight line).

.. much more is OK > OH.

§ 116

Ax. 8

$ 52

Q. E. D.

PROPOSITION IX. THEOREM.

186. A straight line perpendicular to a radius at its extremity is a tangent to the circle.

Let BA be the radius, and MO the straight line perpendicular to BA at A.

We are to prove MO tangent to the circle.

From B draw any other line to M Ó, as BC H.

BH > BA,

(a measures the shortest distance from a point to a straight line).

.. point H is without the circumference.

But BH is any other line than B A,

§ 52

.. every point of the line MO is without the circumference, except A.

.. MO is a tangent to the circle at A.

§ 171

Q. E. D.

187. COROLLARY. When a straight line is tangent to a circle, it is perpendicular to the radius drawn to the point of contact, and therefore a perpendicular to a tangent at the point of contact passes through the centre of the circle.

PROPOSITION X. THEOREM.

188. When two circumferences intersect each other, the line which joins their centres is perpendicular to their common chord at its middle point.

A

C

Let C and C be the centres of two circumferences which intersect at A and B. Let A B be their common chord, and CC' join their centres.

We are to prove C C'L to A B at its middle point.

A

drawn through the middle of the chord AB passes through the centres C and C', $ 184 (a erected at the middle of a chord passes through the centre of the O). .. the line C C', having two points in common with this, must coincide with it.

.. CC is to A B at its middle point.

Q. E. D.

Ex. 1. Show that, of all straight lines drawn from a point without a circle to the circumference, the least is that which, when produced, passes through the centre.

Ex. 2. Show that, of all straight lines drawn from a point within or without a circle to the circumference, the greatest is that which meets the circumference after passing through the

centre.