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the minimum of f(P) in a quadrilateral

Source: China second round 2008 p1

March 3, 2012

geometrycircumcirclegeometry proposed

Problem Statement

Given a convex quadrilateral with

\angle B+\angle D<180

.Let

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

of

ABC

,such that

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

.Knowing that

DA,DC

are tangent to circle

O

,

AC=\sqrt 2

,find the minimum of

f(P)

.

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