Question 105
As shown in Fig. 1, $ABCD$ is a convex quadrilateral with $\angle B + \angle D \lt 180^\circ$, and $P$ is a moving point on the plane. Let
$$
f(P) = PA \times BC + PD \times CA + PC \times AB.
$$
(1) Prove that $P$, $A$, $B$, $C$ are concyclic when $f(P)$ reaches the minimum.
(2) Suppose that point $E$ is on the arc $\widearc{AB}$ of the circumscribed circle $O$ of $\triangle ABC$, satisfying
$$
\frac{AE}{AB} = \frac{\sqrt{3}}{2}, \quad \frac{BC}{EC} = \sqrt{3} - 1, \quad \angle ECB = \frac{1}{2} \angle ECA;
$$
furthermore, $DA$, $DC$ are tangent to $\odot O$, $AC = \sqrt{2}$. Find the minimum of $f(P)$.
Fig. 1
$$
f(P) = PA \times BC + PD \times CA + PC \times AB.
$$
(1) Prove that $P$, $A$, $B$, $C$ are concyclic when $f(P)$ reaches the minimum.
(2) Suppose that point $E$ is on the arc $\widearc{AB}$ of the circumscribed circle $O$ of $\triangle ABC$, satisfying
$$
\frac{AE}{AB} = \frac{\sqrt{3}}{2}, \quad \frac{BC}{EC} = \sqrt{3} - 1, \quad \angle ECB = \frac{1}{2} \angle ECA;
$$
furthermore, $DA$, $DC$ are tangent to $\odot O$, $AC = \sqrt{2}$. Find the minimum of $f(P)$.
Fig. 1