Chapter V: Part II: Axioms and Theorems (3)

Then, because the angle F A E is measured by half the arc F E, and the
angle B A E is measured by half the arc B E,

The difference of the angles, or F A B, must be measured by half the
difference of the arcs, or half of F B.

PROPOSITION XXI. THEOREM.

DEMONSTRATION.

We wish to prove that

_Parallel chords intercept equal arcs of the circumference._

Let the chords A B, C D, be parallel; then will the intercepted arcs A
C and B D be equal.

For draw the straight line B C.

Because the lines A B and C D are parallel, the interior alternate
angles A B C, B C D, are equal.

But the angle A B C is measured by half the arc A C;

And the angle B C D is measured by half the arc B D:

Then, because the angles are equal, the half arcs which measure them
must be equal, and the whole arcs themselves must be equal.

PROPOSITION XXII. THEOREM.

DEMONSTRATION.

We wish to prove that

_The angle formed by a tangent and a chord meeting at the point of
contact is measured by half the intercepted arc._

Let the tangent C A B and the chord A D meet at the point of contact
A; then will the angle B A D be measured by half the intercepted arc
A D.

For draw the diameter A E F.

Because A B is a tangent, and A E a radius at the point of contact,
the angle B A F is a right angle, and is measured by the semicircle
A D F.

Because the angle F A D is at the circumference, it is measured by
half the arc D F.

Then the difference between the angles B A F and D A F, or B A D, must
be measured by half the difference of the arcs A D F and D F, or A
D;

That is, the angle B A D is measured by half the arc A D.

PROPOSITION XXIII. THEOREM.

DEMONSTRATION.

We wish to prove that

_A tangent and chord parallel to it intercept equal arcs of the
circumference._

Let A B be tangent to the circumference at the point D, and let C F be
a chord parallel to the tangent; then will the intercepted arcs C D
and D F be equal.

For from the point of contact D, draw the straight line D C.

Because the tangent and chord are parallel, the interior alternate
angles A D C and D C F are equal.

But the angle A D C, being formed by the tangent D A and the chord D
C, is measured by half the intercepted arc D C;

And the angle D 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.

PROPOSITION XXIV. THEOREM.

DEMONSTRATION.

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 chords A B and C D intersect each other in the point E; then
will the angle B E D or A E C be measured by half the sum of the
arcs A C, B D.

For from the point C draw C F parallel to A B.

Because the chords A B and C F are parallel, the arcs A C, B F, are
equal.

Add each of these equals to B D, and we have B D plus A C equal to B D
plus B F; that is, the sum of the arcs B D, A C, is equal to the arc
F D.

Because the chords A B, C F, are parallel, the opposite exterior and
interior angles D E B, D C F, are equal.

But D C F is an angle at the circumference, and is therefore measured
by half the arc F D.

Then the equal angle D E B must be measured by half of the arc F D, or
its equal B D, plus A C.

PROPOSITION XXV. THEOREM.

DEMONSTRATION.

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 secants A B, A C, intersect the circumference in the points D
and E; then will the angle B A C be measured by half the difference
between the arcs B C and D E.

For from the point D draw the chord D F parallel to E C.

Because A C and D F are parallel, the opposite exterior and interior
angles B D F and B A C are equal.

Because the chords D F, E C, are parallel, the arcs D E and F C are
equal.

If from the arc B C we take the arc D E, or its equal F C, we shall
have left the arc B F;

But the angle B D F, being at the circumference, is measured by half
the arc B F:

Then the equal of B D F, or B A C, must be measured by half the arc B
F, or half the difference between the intercepted arcs B C and D E.