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 $\mathrm{BE}.$

Let $\mathrm{P}$ be the centre of the square constructed on hypotenuse $\mathrm{AC}$ of the right angle triangle $\mathrm{ABC}.$ If $\angle \mathrm{BCA}-\angle \mathrm{BAC}=20°.$ Find the value of $\angle \mathrm{ABP}-\angle \mathrm{CBP}.$

In a trapezium $\mathrm{ABCD},$ where $\mathrm{AB}\Vert \mathrm{CD}$ and diagonals $\mathrm{AC}$ intersects the other diagonal $\mathrm{BD}$ at $\mathrm{O}.$ Which of the following statements are always correct?
(i) $∆\mathrm{AOB}~∆\mathrm{COD}$
(ii) $∆\mathrm{DOA}~∆\mathrm{COB}$
(iii) Area $∆\mathrm{DOA}=$ Area $∆\mathrm{COB}$
(iv) Area $∆\mathrm{AOB}=$ Area $∆\mathrm{COD}$