G Tewani Solutions for Chapter: Solutions and Properties of Triangle, Exercise 3: DPP

$AD, BE, CF$ are internal angular bisectors of $∆ABC$ and $I$ is the incentre. If $a(b+c)\sec \frac{A}{2}ID+b(a+c)\sec \frac{B}{2}IE+c(a+b)\sec \frac{C}{2}IF=kabc,$ then the value of $k$ is:

In $∆ABC$, it is given that the distance between the circumcentre $\left(O\right)$ and orthocentre $\left(H\right)$ is $R\sqrt{1-8\cos A\cos B\cos C}.$ If $Q$ is the mid-point of $OH$, then $AQ$ is

In any $∆ABC,$ the line joining the circumcentre $\left(O\right)$ and incentre $\left(I\right)$ is parallel to $AC,$ then $OI$ is equal to

Let the incircle of a $∆ABC$ touch the sides $BC, CA$ and $AB$ at $D, E$ and $F$, respectively. Let the area of $∆ABC$ be $△$ and that of $DEF$ be ${∆}^{\text{'}}.$ If $a, b$ and $c$ are sides of $∆ABC,$ then the value of $abc(a+b+c)\frac{{∆}^{\text{'}}}{{∆}^3}$ is

Let $H$ be the orthocentre of triangle $ABC.$ Then the angle subtended by side $BC$ at the centre of incircle of $∆CHB$ is:

Let $P$ be an interior point of $∆ABC.$ If $\mathit{\angle }A=45°, \mathit{\angle }B=60°$ and $\mathit{\angle }C=75°$. If $X =$ area of $∆PBC,$ $Y =$ area of $∆PAC$ and $Z =$ area of $∆PAB,$ then which of the following is/are true?

Let $l$ be the length of median from the vertex $A$ to the side $BC$ of a $∆ABC.$ Then:

A circle having centre as $O\text{'}$ and radius $r\text{'}$ touches the incircle of $∆ABC$ externally at $F,$ where $F$ is on $BC$ and also touches its circumcircle internally at $G.$ If $O$ is the circumcentre of $∆ABC$ and $I$ is its incentre, then: