The sides $\mathrm{AC}, \mathrm{BC}$ and $\mathrm{AB}$ of a triangle $\mathrm{ABC}$ are tangents to its in the circle at $\mathrm{D}, \mathrm{E}$ and $\mathrm{F},$ respectively. The point $D$ lies on the hypotenuse $\mathrm{AC}$ such that $\mathrm{D}$ is the circumcentre of the triangle $\mathrm{ABC}.$ Find $\mathrm{EF}: \mathrm{AC}.$

In an isosceles triangle $\mathrm{ABC},$ side $\mathrm{BC}$ is distinct from rest of the two sides. $\mathrm{CD}$ and $\mathrm{BE}$ are the medians on $\mathrm{AB}$ and $\mathrm{AC},$ respectively. If $\mathrm{W},\mathrm{X},\mathrm{Y}$ and $\mathrm{Z}$ represents distinct areas in the triangle, find the ratio of $\mathrm{W}: \mathrm{X}: \mathrm{Y}: \mathrm{Z}.$

In the following isosceles triangle, in which $\mathrm{AC}=\mathrm{BC}$ and $\mathrm{AB}=\mathrm{BD}=\mathrm{DE}=\mathrm{EF}=\mathrm{FC}=6 \mathrm{cm},$ find $\mathrm{BE}.$