In a right-angled triangle, the bisector of the right angle divides the hypotenuse in the ratio $m:n$. If in $\frac{m^k}{n^k}$ ratio the hypotenuse is divided by the altitude dropped from the vertex of the right angle then find $k$.

$ABCD$ is a parallelogram. $AB$ is divided at $P$ and $CD$ at $Q$, so that $AP:BP=3:2$ and $CQ:QD=4:1$. If $PQ$ meets $AC$ at $R$, then prove that $AR=\frac{3}{7}AC$.

If $I$ is the incentre of $△ABC$ and $AI$ meets $BC$ at $D$, prove that $\frac{AI}{ID}=\frac{b+c}{a}$, where $a, b, c$ are lengths of sides $BC, CA, AB$ of $△ABC$ respectively.

In $△ABC$, the bisector of $\angle B$ intersects the side $AC$ at $D$. A line parallel to side $AC$ intersects line-segments $AB, DB$ and $CB$ at points $P, R$ and $Q$, respectively. Prove that $AB\times CQ=BC\times AP$

In $△ABC$, the bisector of $\angle B$ intersects the side $AC$ at $D$. A line parallel to side $AC$ intersects line-segments $AB, DB$ and $CB$ at points $P, R$ and $Q$, respectively. Prove that $PR\times BQ=QR\times BP$

$BO$ and $CO$ are respectively the bisectors of $\angle B$ and $\angle C$ of $△ABC$. $AO$ produced meets $BC$ at $P$. Show that $\frac{AB}{BP}=\frac{AO}{OP}$.

$BO$ and $CO$ are respectively the bisectors of $\angle B$ and $\angle C$ of $∆ABC.$ $AO$ produced meets $BC$ at $P.$ Show that $\frac{AC}{CP}=\frac{AO}{OP}.$

$BO$ and $CO$ are respectively the bisectors of $\angle B$ and $\angle C$ of $△ABC$. $AO$ produced meets $BC$ at $P$. Show that $AP$ is the bisector of $\angle BAC$.