1

Part of 2008 China Second Round Olympiad

Problems(1)

the minimum of f(P) in a quadrilateral

Source: China second round 2008 p1

Given a convex quadrilateral with

\angle B+\angle D<180

P

be an arbitrary point on the plane,define

f(P)=PA*BC+PD*CA+PC*AB

. (1)Prove that

P,A,B,C

are concyclic when

f(P)

attains its minimum. (2)Suppose that

E

is a point on the minor arc

AB

of the circumcircle

O

ABC

AE=\frac{\sqrt 3}{2}AB,BC=(\sqrt 3-1)EC,\angle ECA=2\angle ECB

DA,DC

are tangent to circle

O

AC=\sqrt 2

,find the minimum of

f(P)

geometrycircumcirclegeometry proposed