We wish to prove that
A tangent and chord parallel to it intercept equal arcs of the circumference.
LetA Bbe tangent to the circumference at the pointD, and letC Fbe a chord parallel to the tangent; then will the intercepted arcsC DandD Fbe equal.
For from the point of contactD, draw the straight lineD C.
Because the tangent and chord are parallel, the interior alternate anglesA D CandD C Fare equal.
But the angleA D C, being formed by the tangentD Aand the chordD C, is measured by half the intercepted arcD C;
And the angleD C F, being at the circumference, is measured by half the arc on which it stands,D F:
Then, because the angles are equal, the half arcs which measure them are equal, and the arcs themselves are equal.
We wish to prove that
The angle formed by the intersection of two chords in a circle is measured by half the sum of the intercepted arcs.
Let the chordsA BandC Dintersect each other in the pointE; then will the angleB E DorA E Cbe measured by half the sum of the arcsA C,B D.
For from the pointCdrawC Fparallel toA B.
Because the chordsA BandC Fare parallel, the arcsA C,B F, are equal.
Add each of these equals toB D, and we haveB DplusA Cequal toB DplusB F; that is, the sum of the arcsB D,A C, is equal to the arcF D.
Because the chordsA B,C F, are parallel, the opposite exterior and interior anglesD E B,D C F, are equal.
ButD C Fis an angle at the circumference, and is therefore measured by half the arcF D.
Then the equal angleD E Bmust be measured by half of the arcF D, or its equalB D, plusA C.
We wish to prove that
The angle formed by two secants meeting without a circle is measured by half the difference of the intercepted arcs.
Let the secantsA B,A C, intersect the circumference in the pointsDandE; then will the angleB A Cbe measured by half the difference between the arcsB CandD E.
For from the pointDdraw the chordD Fparallel toE C.
BecauseA CandD Fare parallel, the opposite exterior and interior anglesB D FandB A Care equal.
Because the chordsD F,E C, are parallel, the arcsD EandF Care equal.
If from the arcB Cwe take the arcD E, or its equalF C, we shall have left the arcB F;
But the angleB D F, being at the circumference, is measured by half the arcB F:
Then the equal ofB D F, orB A C, must be measured by half the arcB F, or half the difference between the intercepted arcsB CandD E.