Kesab Chandra Nag Solutions for Exercise 1: EXERCISE-1

$AD,BE$ and $CF$ are three medians of the $∆ABC$. Prove that $AD+BE>CF$.

$O$ is the ex-centre of the $∆ABC$. Prove that $\angle BOC=90°-\frac{1}{2}\angle BAC$.

Two medians $BE$ and $CF$ of $∆ABC$ intersect at $G$. Prove that area of the quadrilateral $AFGE=\frac{1}{2}$ area of $∆ABC$.

$I$ is the incentre of $∆ABC$. $AI$ bisects $BC$ at $D$. Prove that $∆ABC$ is isosceles.

Two medians $BE$ and $CQ$ of $∆ABC$ intersects at $G$. Prove that $∆BGQ=\frac{1}{6}∆ABC$.

$I$ is the incentre of the $∆ABC$. The perpendicular drawn from $I$ to $BC$ intersects $BC$ at $P$. Prove that $AB-AC=BP-CP$.

The interior bisector of $\angle B$ and the exterior bisector of $\angle C$ of the $∆ABC$ intersects each other at $O$. Prove that $\angle BOC=\frac{1}{2}\angle A$.

$P$ is a point on the circum-centre of the $△ABC$. $PD, PE$ and $PF$ are the perpendiculars drawn from $P$ to $AB, BC$ and $CA$. Check whether $D, E$ and $F$ are collinear.