MathDB

Points

A

and

B

are given on a line having no common points with a sphere

K

. The feet

P

of the perpendicular from the center of

K

to the line

AB

is positioned between

A

and

B

, and the lengths of segments

AP

and

BP

both exceed the radius of

K

. Consider the set

Z

of all triangles

ABC

whose sides

AC

and

BC

are tangent to

K

. Prove that among all triangles in

Z

, a triangle

T

with a maximum perimeter also has a maximum area.

Problem Statement