Amit M Agarwal Solutions for Chapter: Properties and Solutions of Triangles, Exercise 3: Objective Questions

In an isosceles right $\Delta ABC,$ the value of $\frac{IP}{OP}$ is, (where $P$ is orthocentre, $I$ is incentre, $O$ is circumcentre)

Consider a triangle $ABC.$ $C{C}_1$ and $B{B}_1$ are the medians drawn through angular points $B$ and $C$, respectively and ${}^{\text{'}}G^{\text{'}}$ is the centroid of $\Delta ABC.$ If the points $A, C_1, G$ and $B_1$ are concyclic, then 

If $a, b, c$ are sides of the $\text{Δ}ABC,$ such that ${\left(1+\frac{b-c}{a}\right)}^a·{\left(1+\frac{c-a}{b}\right)}^b·{\left(1+\frac{a-b}{c}\right)}^c\ge 1$, then triangle $ABC$ must be 

In a $\Delta$ with sides $a, b$ and $c$, a semicircle touching the sides $AC$ and $CB$ is inscribed whose diameter lies on $AB.$ Then the radius of the semicircle is 

In a $\Delta ABC,\angle B=\frac{\pi }{2},$ then value of radius of incircle is:

In a $\Delta ABC$, the point $D$ divides $BC$ in the ratio $1: 2$. Also, $AD$ is perpendicular to $AB.$ Then the value of the expression $\tan B(1+2\tan A\tan C)-2\tan C$ is

If $\sin A$ and $\sin B$ of a triangle $ABC$ satisfy $c^2{x}^2-c\left(a+b\right)x+ab=0,$ then the triangle is:

With usual notations, consider a $\Delta ABC$, with given $\angle A$ and side $a$. If $bc=x^2$, then such a triangle would exist, if ($x$ is a given positive real number)