MathDB

Let

C

be a closed convex curve with a continuously turning tangent and let

O

be a point inside

C.

For each point

P

C

we define

T(P)

as follows: Draw the tangent to

C

P

and from

O

drop the perpendicular to that tangent. Then

T(P)

is the point at which

C

intersects this perpendicular. Starting now with a point

P_{0}

C

, define points

P_n

P_n =T(P_{n-1}).

Prove that the points

P_{n}

approach a limit and characterize all possible limit points. (You may assume that

T

is continuous.)

Problem Statement