A is a cevian that divides the opposite side into two congruent lengths.

In the triangle $ABC$, $BD$ is a cevian. The area of $△BCD$ is $40 {\mathrm{cm}}^2$. Find the area of $△ABD$,

In the triangle $ABC$, $BD$ is a cevian. The area of $△ABD$ is $21 {\mathrm{cm}}^2$. Find the area of $△BCD$,

In the triangle $ABC$, $BD$ is a cevian. The area of $△BCD$ is $8 {\mathrm{cm}}^2$ and $△ABD$ is$10 {\mathrm{cm}}^2$. If the length of $AD$ is $5 \mathrm{cm}$. Find $CD$.

In triangle $ABC$, points $D, E, F$ are on the side lines $CA, AB, BC$ respectively and the points $D, E, F$ are collinear. Find $AD$.

The cevians $AF, BG$ and $CE$ of the triangle intersect at $D$. $AG=2 \mathrm{cm}, BF=3 \mathrm{cm}, CF=9 \mathrm{cm}, AE=4 \mathrm{cm}$ and $BE=4 \mathrm{cm}$. Find $GC$.

The cevians $AF, BG$ and $CE$ of the triangle intersect at $D$. $AG=2 \mathrm{cm}, GC=6 \mathrm{cm}, CF=9 \mathrm{cm}, AE=4 \mathrm{cm}$ and $BE=4 \mathrm{cm}$. Find $BF$.

In triangle $ABC$, points $D, E, F$ are on the side lines $CA, AB, BC$ respectively and the points $D, E, F$ are collinear. Find $CD$.

The cevians $AF, BG$ and $CE$ of the triangle intersect at $D$. $AG=2 \mathrm{cm}, GC=6 \mathrm{cm}, CF=9 \mathrm{cm}, BF=3 \mathrm{cm}$ and $BE=4 \mathrm{cm}$. Find $AE$.

In triangle $ABC$, points $D, E, F$ are on the side lines $CA, AB, BC$ respectively and the points $D, E, F$ are collinear. Find $BC$.

In a garden containing several trees, three particular trees $P, Q, R$ are located in the following way, $BP=2 \mathrm{m}, CQ=3 \mathrm{m}, RA=10 \mathrm{m}, PC=6 \mathrm{m}, QA=5 \mathrm{m}, RB=2 \mathrm{m}$ where $A, B, C$ are points such that $P$ lies on $BC, Q$ lies on $AC$ and $R$ lies on $AB$. Check whether the trees $P, Q, R$ lie on a same straight line.

Prove Ceva's theorem.

In the given figure, $ABC$ is a triangle with $\angle B={90}^{\mathrm{o}}$, $BC=3 \mathrm{cm}$ and $AB=4 \mathrm{cm}$. $D$ is point on $AC$ such that $AD=1 \mathrm{cm}$ and $E$ is the midpoint of $AB$. Join $D$ and $E$ and extend $DE$ to meet $CB$ at $F$. Find $BF$.