In a triangle ABC, the side AB is bisected at the point D and the line segment CD is bisected at the point E. If AE produced intersects BC at the point. F, then prove that$FC=\frac{1}{3}BC$.

In figure, $\angle D=\angle E$ and $\frac{AD}{DB}=\frac{AE}{EC}$, prove that $BAC$ is an isosceles triangle.

In the given figure $DE \left|\right| BC$ and $\frac{AD}{DB}=\frac{3}{5}$ if $AC=16$ units then the value of $AE$ will be

In $∆ABC, DE\parallel BC,$ If $AD=5 cm, BD=7 cm$ and $AC=18 cm,$ If the length of $AE=k$, then find the value of $k$.

In $∆\mathrm{ABC}, \mathrm{LM}\parallel \mathrm{BC}$ and $\frac{\mathrm{AL}}{\mathrm{LB}}=\frac{2}{3}, \mathrm{AM}=5 \mathrm{cm}$, find $\mathrm{AC}$

In Fig 2, $DE\parallel AC$ and $DC\parallel AP$. Prove that $\frac{BE}{EC}=\frac{BC}{CP}$

In $△ABC,$ the line parallel to $BC$ meets $AB$ and $AC$ at $P$ and $Q$. If $AP=8 \mathrm{cm}, CQ=9 \mathrm{cm}$ and $AQ=2BP$, then what is the value of $BP$?

In $∆ABC, O$ is a point on the median $AD. BO$ and $CO$ produced meet $AC$ and $AB$ at $G \& F$. Prove that $FG\parallel BC$.

Two parallel straight lines intersect three concurrent straight lines at $A,B,C$ and $X,Y,Z$. Prove that $AB:BC=XY:YZ$.

In trapezium $ABCD$, $AB$ and $DC$ are parallel. A straight line parallel to $AB$ meets $AD$ and $BC$ at $E$ and $F$. Prove that $DE:EA=CF:FB$.

In the figure,$\mathrm{ABC}$ is a triangle in which $\mathrm{AB}=\mathrm{AC}$ . Points$\mathrm{D}$ and $\mathrm{E}$ are points on the side $\mathrm{AB}$ and $\mathrm{AC}$ respectively such that $\mathrm{AD}=\mathrm{AE}$. Show that the points $\mathrm{B}, \mathrm{C}, \mathrm{E}$ and $\mathrm{D}$ lie on a same circle.

In $△ ABC, DE\parallel DE\parallel BC$. If $AD=5 \mathrm{cm}, BD=7 \mathrm{cm}$ and $AC=18 \mathrm{cm}$, find the length of $AE$ in centimrter.

In $∆ABC$ the line parallel to $BC$ meets $AB$ and $AC$ at $P$ and $Q$ and $\frac{AP}{PB}=\frac{2}{5}$. If $AC=28 \mathrm{cm}$, then length of $AQ$ is 

In $△ABC,$ the line parallel to $BC$ meets $AB$ and $AC$ at $D$ and $E$. If $AD:DB=3:2$ and $AC=10 \mathrm{cm}$, thus what is the value of $AE$?

In trapezium $ABCD, AB\parallel CD$; The diagonals $AC \& BD$ meet at $O$. Prove that $AO\times OD=BO\times OC$.

In $△ABC,$ the line parallel to $BC$ meets $AB$ and $AC$ at $P$ and $Q$. If $AB=3PB$, then find $PQ:BC$.

In $∆ABC$ the line parallel to $BC$ meets $AB$ and $AC$ at $X$ and $Y$ and $AX:XB=3:1$. If $AY=6.6 \mathrm{cm}$, then length of $AC$ is

In $∆DEW, AB \left|\right| EW$. If $AD = 4 \mathrm{cm}$, $DE = 12 \mathrm{cm}$ and $DW = 24 \mathrm{cm}$, then find the value of $DB$.

In $∆\mathrm{ABC}$ the line parallel to $\mathrm{BC}$ meets $\mathrm{AB}$ and $\mathrm{AC}$ at $\mathrm{X}$ and $\mathrm{Y}$. If $\mathrm{BX}=\mathrm{AY}, \mathrm{AX}=6.4$ unit, $\mathrm{CY}=8.1$ unit then length of $\mathrm{AY}$ is