Question

In Fig., $PS$ is the bisector of $\angle QPR$ of $\Delta PQR.$
Prove that $\frac{QS}{SR}=\frac{PQ}{PR}.$

Answer 1

Draw a line parallel to $PS$ through $R$
It intersects $QP$ produced at $T$.

By using angle bisector properties
In $\Delta QPR$
$\angle QPS=\angle SPR$ ($PS$ is bisector of $\angle QPR$) .....(1)

By using alternate interior angle property For parallel lines $PS$ and $RT$ and transversal $PR$
$\angle PRT=\angle SPR$ (Alternate interior angles) ..........(2)

By using corresponding angle property
For parallel lines $PS$ and $RT$ and transversal $QT$
$\angle QPS=\angle PTR$ (Corresponding angles) ........(3)
From (1), (2) and (3)
$\Rightarrow \angle PTR=\angle PRT$

Sides opposite to the equal angles are equal.
$\Rightarrow PR=PT........\left(4\right)$

Apply $BPT$ theorem in $\Delta QTR$
In $\Delta QTR$
$PS||RT$
$\Rightarrow \frac{QS}{SR}=\frac{QP}{PT}$
From equation (4)
$\therefore \frac{QS}{SR}=\frac{QP}{PR}$
Hence, proved.