Mathematics

Problem

Let

T

be a point of contact of a tangent drawn from an external point

P

to a circle with centre

O

. The second tangent from

P

touches the circle at

S

. The points

T

and

S

are joined to form chord

TS

, and

O

is joined to

T

and

S

. a. Prove that

\triangle OTP \cong \triangle OSP

. b. Deduce that

OT = OS

and

\angle TOP = \angle POS

. c. Prove that quadrilateral

TOSP

is cyclic. d. If the radius of the circle is

r

and

\angle TOS = \theta

, express the area of

\triangle TPS

in terms of

r

and

\theta