A

II. CONVERSELY. Given in equal ©, 0 and 0',

Ex. 523. If ABC is any triangle inscribed in a circle and ZA>LB,

then BC > AC.

Ex. 524.

State and prove the converse of the preceding proposition.

Ex. 525. In the diagram opposite, if

Ex. 526. In the same diagram, if AD = DC, and

AB> BC, then Z ADB><BDC.

Ex. 527. If upon a radius two perpendicular chords are drawn, the one nearer the center is the greater chord.

Ex. 528. In the diagram opposite, if AB> CD, and arcs ACB and DAC are minor arcs, prove that CB > AD.

Ex. 529. In the same diagram, if CB>AD, and arcs DAC and ACB are minor arcs, prove that ABCD

D

PROPOSITION VII

200. In the same circle, or in equal circles, chords unequally distant from the center are unequal, the nearer one being the greater, and conversely.

H A

Given in O ABDC, OE 1 chord AB, OF chord CD, and

201. COR. The diameter is greater than any other chord.

202. METHOD XVI. The inequality of chords, in the same circle or in equal circles, is usually established by means of unequal distances from the center or by means of unequal arcs.

Ex. 530. The perpendicular from the center of a circle to a side of an inscribed equilateral hexagon is less than the perpendicular from the center to the side of an inscribed equilateral octagon.

Ex. 531. In the diagram opposite, if E01 AB, OF CD, and ▲ OEF>ZOFE, then AB > CD.

Ex. 532. If AABC is inscribed in a circle and <B>C, then the perpendicular from the center upon AB is greater than the perpendicular from the center upon AC.

B

Ex. 533. The shortest chord which can be drawn through a point within a circle is perpendicular to the radius drawn through the point. Ex. 534. Two chords drawn from a point in the circumference are unequal if they make unequal angles with the radius drawn from that point. * Ex. 535. Two chords drawn through an interior point are unequal if they make unequal angles with the radius drawn through that point.

PROPOSITION VIII. THEOREM

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

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

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

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

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

208. DEF. The length of tangent drawn from a point to a circle is the length of the line from this point to the point of contact; as AB (Prop. IX).

PROPOSITION IX. THEOREM

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

HINT. What is the means of proving the equality of lines?

210. COR. The tangents drawn to a circle from a point without make equal angles with the line joining the point to the center (i.e≤ OAB = ≤ OAC).

211. DEF. A common tangent to two circles is called an internal

tangent if it lies between the two circles, as AB.

Otherwise it is

A

called a common external tangent,

as CD.

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

212. DEF. A polygon is circumscribed

about a circle if all its sides are tangent to E

the circle; as ABCDE. The circle is then

said to be inscribed in the polygon.

The center of the inscribed circle is called the incenter of the polygon.

B

Ex. 536. The common internal tangents of two circles are equal. Ex. 537. The common external tangents of two circles are equal. Ex. 538. A chord forms equal angles with the tangents drawn at its ends.

Ex. 539. The sum of two opposite sides of a circumscribed quadrilateral is equal to the sum of the other two sides.

Ex. 540. If hexagon ABCDEF is circumscribed about a circle, AB+ CD + EF = BC + DE + FA.

Ex. 541.

The sum of the arms of a right triangle circumscribed about a circle is equal to the hypotenuse increased by the diameter of the circle. Ex. 542. If two tangents make an angle of 60°, the chord joining the points of contact equals either tangent.

*Ex. 543. Triangle ABC is circumscribed about a circle, which touches the sides AB, BC, and CA in X, Y, and Z, respectively. Find the lengths of AX, BY, and CZ if AB = 3, BC = 4, and CA = 5.

[See practical applications, p. 291.]

213. DEF. The line of centers is the line joining the centers of two circles.

Thus 00' (diagram for Prop. X) is the line of centers of circles O and O'.