Exercise

Construct a $△ABC$ with base $BC=3 \mathrm{cm}$, vertical angle $\angle BAC=45˚$  and median through the vertex $A$ is $3.5 \mathrm{cm}$.

Construct a $△ABC$ with base $BC=4 \mathrm{cm}$, vertical angle $\angle BAC=40˚$  and median through the vertex $A$ is $4.5 \mathrm{cm}$.

Construct a $△ABC$ such that $AB=5 \mathrm{cm}, \angle C=45°$, and the altitude from $C$ to $AB$ is of length $2 \mathrm{cm}$.

Construct a $△ABC$ such that $AB=6 \mathrm{cm}, \angle C=45°$, and the altitude from $C$ to $AB$ is of length $2.5 \mathrm{cm}$.

Construct a $△ABC$ with base $BC=6 \mathrm{cm}$, vertical angle $\angle BAC=60˚$  and median through the vertex $A$ is $4 \mathrm{cm}$.

Prove Ceva's theorem.

Prove that the external angle bisectors of a triangle intersect their opposite sides at three collinear points.

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 $BF$.