You are given a triangle ABC, a point D on segment AC, a point E on segment AB and a point F on segment BC. Let BD and CE intersect in point P. Join P with F. Suppose that $\angle\mathrm{EPB} = \angle\mathrm{BPF} = \angle\mathrm{FPC} = \angle\mathrm{CPD}$ and $\mathrm{PD} = \mathrm{PE} = \mathrm{PF}$.
For each statement below, state if it is true or false.
(i) AP must bisect $\angle\mathrm{BAC}$.
(ii) $\triangle\mathrm{ABC}$ must be isosceles.
(iii) $\mathrm{A}$, $\mathrm{P}$, $\mathrm{F}$ must be collinear.
(iv) $\angle\mathrm{BAC}$ must be $60^{\circ}$.