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PROPOSITION VI.

THEOREM. The angle formed by a tangent and chord is measured by half the arc of that chord.

Let AB be a tangent, and CD a chord drawn from the point of contact C; then the angle BCD will be measured by half the arc CGD, and the angle ACD will be measured by half the arc CLD.

A

G

Draw the radius FC to the point of contact, and the radius FG perpendicular to the chord CD, meeting it at the point K. Then the radius FG, being perpendicular to the chord CD, bisects the arc CGD, (B. III, Prop. 1;) therefore CG is half the arc CGD.

In the triangle CFK, the angle CKF being a rightangle, the sum of the two remaining angles CFK and FCK is equal to a right-angle, (B. I, Prop. xxiv, Cor.,) which is consequently equal to the right-angle FCB. From each of these equals, take away the common angle FCK, and there remains the angle CFK equal to the angle BCD. But the angle CFK is measured by the arc CG, (B. III, Def. 13,) which is half of the arc CGD; therefore, the equal angle BCD must also be measured by half the arc CGD.

Again, the line LFG, being perpendicular to the chord CD, bisects the arc CLD, (B. III, Prop. 1;) therefore the arc CL is half the arc CLD. Now, the line CF, meet

ing GL, makes the sum of the two angles CFG, CFL equal to two right-angles, (B. I, Prop. 1,) and the line CD makes with AB the two angles DCB, DCA together equal to two right-angles. If from these equals we take the equals CFG, DCB, we shall have the remainders CFL, DCA equal. Now, the former of these, CFL, being an angle at the centre, is measured by the arc CL, which is half the arc CLD; therefore, the angle DCA is also measured by half the arc CLD.

Cor. 1. The sum of two right-angles is measured by half the circumference; for the two angles BCD, ACD, which together make two right-angles, are respectively measured by the arcs CG, CL, which make half the circumference, LG being a diameter.

Cor. 2. Hence, also, one right-angle must have for its measure a quarter of the circumference.

PROPOSITION VII.

THEOREM. An angle at the circumference of a circle is measured by half the arc that subtends it.

Let BAC be an angle at the circumference: it has for its measure half the arc BC, which subtends it.

For, suppose the tangent DF to pass through the angular point A. Then, the angle DAC being measured by half the arc half the arc AB, (B. III

D

B

D

ABC, and the angle DAB by
Prop. vI,) it follows that the

difference of those angles is measured by half the difference of the said arcs; that is, the angle BAC is meas'ured by half the arc BC upon which it stands.

Cor. 1. All angles in the same segment of a circle, or subtended by the same arc, are equal to each other; for each is measured by half the same arc.

Cor. 2. An angle at the centre of a circle is double the angle at the circumference, when both are subtended by the same arc. For the angle at the centre is (B. III, Def. 13,) measured by the whole arc on which it stands, and the angle at the circumference is measured by half the arc on which it stands; consequently the angle at the centre is double the angle at the circumference.

Cor. 3. An angle in a semicircle is a right-angle; for it is measured by half a semicircumference, or by a quadrant, which is the measure of a right-angle, (B. III, Prop. vi, Cor. 2.)

Cor. 4. The sum of any two opposite angles of a quadrilateral inscribed in a circle, is equal to two rightangles; for, as each inscribed angle is measured by half the arc which subtends it, it follows that the two opposite angles of an inscribed quadrilateral, together, must be measured by half the entire circumference, which is the measure of two right-angles.

PROPOSITION VIII.

THEOREM. The angle formed by a tangent to a circle, and a chord drawn from the point of contact, is equal to the angle in the alternate segment.

If AB be a tangent, AC a chord, and D any angle in the alternate segment ADC; then will the angle D be equal to the angle BAC made by the tangent and the chord of the arc AFC.

For the angle D at the circum

ference is measured by half the

D

B

arc AFC, (B. III, Prop. vii,) and the angle BAC made by the tangent and chord is also measured by half the same arc AFC, (B. III, Prop. vII;) therefore these two angles are equal.

PROPOSITION IX.

THEOREM. If any side of a quadrilateral, inscribed in a circle, be produced, the outward angle will be equal to the inward and opposite angle.

If the side BA of the quadrilateral ABCD, inscribed in a circle, be produced to F, the outward angle DAF will be equal to the inward and opposite angle C.

For, the sum of the two adja

F

A

D

cent angles DAF and DAB is equal to two right-angles, (B. I, Prop. 1,) and the sum of the two opposite angles C and DAB is also equal to two right-angles, (B. III, Prop. vII, Cor. 4;) therefore the two angles DAF and DAB are together equal to the sum of DAB and C. From each, take the common angle DAB, and we have the remainders DAF and C equal to each other.

PROPOSITION X.

THEOREM. Any two parallel chords intercept equal arcs.

Let the chords AB, CD be parallel; then will the arcs AC, BD be equal.

Draw the line BC; then, because AB and CD are parallel, the alternate angles B and C are equal, (B. I, Prop. xvII.) But the

angle B at the circumference is measured by half the arc AC, (B. III, Prop. vII,) and the equal angle C at the circumference is measured by half the arc BD; therefore the arc AC is equal to the arc BD.

THEOREM.

PROPOSITION XI.

When a tangent and chord are parallel to each other, they intercept equal arcs.

Let the tangent ABC be parallel to the chord DF; then will the arc BD equal BF.

Draw the chord BD. Then, since the lines AC, DF are parallel, the alternate angles DBA and BDF are equal (B. I, Prop. xvII.) But the angle DBA, formed by a

B

A

F

tangent and chord, is measured by half the arc BD, (B. III Prop. VII;) and its equal angle BDF, being at the cir cumference, is measured by half the arc BF, (B. III Prop. vII;) therefore the arc BD is equal to BF.

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