Ex. 328. The side of an equilateral hexagon inscribed in a circle is nearer to the center than the side of an equilateral heptagon inscribed in the same circle, and more remote from the center than the side of an equilateral pentagon inscribed in the same circle.

Ex. 329. The shortest chord which can be drawn through a point within a circle is perpendicular to the radius drawn through the point. Which is the longest?

Ex. 330. Two chords drawn from a point in the circumference are unequal if they make unequal angles with the radius drawn from that point. Which of the chords is the greater?

* Ex. 331. Two chords drawn through an interior point are unequal if they make unequal angles with the radius drawn through that point. Which is the greater one?

PROPOSITION VIII. THEOREM

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

192. COR. 1. A tangent is perpendicular to the radius drawn to the point of contact.

193. COR. 2. A perpendicular to a tangent at the point of contact passes through the center of the circle.

194. COR. 3. A perpendicular from the center to a tangent meets it at the point of contact.

195. COR. 4. At a given point of contact there can be one tangent only.

196. The tangents drawn to a circle from a point without are equal.

B

Hyp. In O 0,

AB and AC are tangent.

197. DEF. A common tangent of two circles is called an interior tangent when it lies between the two circles; otherwise, it is called an exterior tangent.

198. DEF. The length of a common tangent is the length of the segment between the points of contact.

Ex. 332. The common internal tangents of two circles are equal. Ex. 333. The common external tangents of two circles are equal. Ex. 334. A chord forms equal angles with the tangents drawn at its ends.

Ex. 335. The sum of two opposite sides of a circumscribed quadrilateral is equal to the sum of the other two opposite sides.

Ex. 336. Find a similar proposition for the circumscribed hexagon. Ex. 337. If two tangents make an angle of 60°, the chord joining the points of contact equals the tangents.

PROPOSITION X. THEOREM

199. If two circumferences intersect, a straight line joining their centers bisects their common chord at right angles.

со

Hyp. Circumferences ACB and ADB intersect at A and B. To prove 00', joining their centers, is the perpendicularbisector of AB.

Proof.

O and O' are each equally distant from A and B.

(Why?)

.. 00' is the perpendicular-bisector of AB. (111) Q.E.D. 200. DEF. The line of centers is the line joining the centers of two circles.

201. DEF. Two circles are tangent to each other if both are tangent to a straight line at the same point. They are tangent internally or externally, according as one circle lies within or without the other.

PROPOSITION XI. THEOREM

202. If two circles are tangent to each other, their line of centers passes through the point of contact.

Hyp. 00' is the line of centers of circles, O and O', tangent at C.

Proof. At the point of contact, C, draw a perpendicular to the common tangent.

This

posses through and O'

Ex. 338. What are the relative positions of two circles, if the li centers is

(a) Greater than the sum of the radii ?

(b) Equal to the sum of the radii ?

(c) Smaller than the sum but greater than the difference of the r (d) Equal to the difference of the radii ?

(e) Smaller than the difference of the radii ?

(J) Equal to zero ?

EXERCISES

Ex. 339. If a secant intersects two concentric circles, its segm intercepted by the two circumferences, are equal.

Ex. 340. Two parallel chords, drawn through the extremities diameter, are equal.

Ex. 341. In the annexed diagram, if the radius OB is equal to prove ▲ COD = 3 Z A.

Ex. 342. If from A the tangents AB and AC are drawn to a circle O, and a third tangent intersects AB and AC in D and F,