Embibe Experts Solutions for Chapter: Properties of Triangle, Exercise 4: EXERCISE-4

In a triangle the angles $A, B, C$ are in A.P. Show that $2\cos \frac{A-C}{2}=\frac{a+c}{\sqrt{a^2-ac+c^2}}$.

In a scalene triangle $ABC$ the altitudes $AD$ $\&CF$ are dropped form the vertices $A \& C$ to the sides $BC \& AB$. The area of $∆ABC$ is known to be equal to $18$, the area of triangle $BDF$ is equal to $2$ and length of segment $DF$ is equal to $2\sqrt{2}$. Find the radius of the circle circumscribing $∆ABC$.

With reference to a given circle, $A_1$ and $B_1$ are the areas of the inscribed and circumscribed regular polygons of $n$ sides, $A_2$ and $B_2$ are corresponding quantities for regular polygons of $2n$ sides : Prove that
(a) $A_2$ is a geometric mean between $A_1$ and $B_1$
(b) $B_2$ is a harmonic mean between $A_2$ and $B_1$

Let $a, b, c$ be the sides of a trianlge & $\Delta$ its area. Prove that $a^2+b^2+c^2\ge 4\sqrt{3}\Delta$, and find when does the equality hold?

If the bisector of angle $C$ of triangle $ABC$ meets $AB$ in $D$ & the circumcircle in $E$, then prove that, $\frac{CE}{DE}=\frac{(a+b{)}^2}{c^2}$.

For any triangle $A B C$, if $B=3C$, show that $\cos C=\sqrt{\frac{b+c}{4c}}$ & $\sin \frac{A}{2}=\frac{b-c}{2c}$.

Let$1<m<3$ . In a triangle $A B C$, if $2 b=(m+1) a$ & $\cos A=\frac{1}{2}\sqrt{\frac{(m-1)(m+3)}{m}}$ prove that there are two values to the third side, one of which is $m$ times the other.

$DEF$ is the triangle formed by joining the points of contact of the incircle with the sides of the triangle $A B C$, prove that
(a) its sides are $2r\cos \frac{A}{2},2r\cos \frac{B}{2}$ and $2r\cos \frac{C}{2}$ where $r$ is the radius of incircle of $∆ABC$.
(b) its angles are $\frac{\pi }{2}-\frac{A}{2},\frac{\pi }{2}-\frac{B}{2}$ and $\frac{\pi }{2}-\frac{C}{2}$
(c) its area is $\frac{r^2 s}{2R}$ where '$s$' is the semiperimeter and $R$ is the circumradius of the $∆ABC$.