THEOREM XV.

211. Any angle formed by a tangent and a chord is measured by half the intercepted arc.

Let the ABD be formed by the tangent CD and the chord AB.

ba L, (195) and is measured by arc EB, (208)

anda is measured by arc AE;

(206)

.. La + ≤ b is measured by (arc AE + arc EB),

or

▲ ABD is measured by arc AEB.

Q. E. D.

THEOREM XVI.

212. Any angle formed by two chords intersecting is measured by half the sum of the arcs intercepted between its sides and the sides of its vertical angle.

Let the a be formed by the chords AB and CD.

To prove that a is measured by (arc AD + arc CB).

La is measured by (arc AD + arc CB).

α

Q. E. D.

THEOREM XVII.

213. Any angle formed by two secants is measured by half the difference of the intercepted arcs.

Let the a be formed by the secants AB and BC.

To

prove that a is measured by (arc AC –

arc EF).

214. SCHOLIUM.-This proposition is true for any position of the secants; hence it is true if one or both become tangents.

With A and B as centres and a radius greater than half of AB, describe arcs intersecting at C and E.

Draw CE

Then

CE bisects AB at the point P. (55) Q.E.F.

PROBLEM 11.

216. At any point in a straight line to erect a perpendicular.

Let P be any point in the straight line AB.

B

D

E

Cut off PD — PE.

With D and E as centres and a radius greater than PD or PE, describe arcs intersecting at C.

Draw CP.

Then

CP is to AB.

(55) Q.E.F.