Name: Distance Between Various Centre Points
Uploaded: 2019-07-19
Duration: 8 min 36 s
Description: This geometry video shows how to identify the location of the incentre, circumcentre, orthocentre and centroid of a triangle. Watch this video and learn more!

Question

If

f, g, h

are internal bisectors of the angles of a triangle

A B C

, show that

\frac{1}{f} \cos \frac{A}{2} + \frac{1}{g} \cos \frac{B}{2} + \frac{1}{h} \cos \frac{C}{2} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}

.

Accepted Answer

Let $A F = f, B G = g$ and $C H = h$ .

Area of $∆ A B C =$ Area of $∆ A B F +$ Area of $∆ A C F$

$\Rightarrow \frac{1}{2} b c \cdot \sin A = \frac{1}{2} \cdot cf \sin (\frac{A}{2}) + \frac{1}{2} \cdot b f \sin (\frac{A}{2})$

$\Rightarrow b c \sin A = cf \sin (\frac{A}{2}) + b f \sin (\frac{A}{2})$

$\Rightarrow 2 b c \sin (\frac{A}{2}) \cos (\frac{A}{2}) = (f c + b f) \sin (\frac{A}{2})$

$\Rightarrow 2 b c \cos (\frac{A}{2}) = f c + b f$

Divide by $f$ we get

$(\frac{2 b c}{f}) \cos (\frac{A}{2}) = b + c$

$\Rightarrow \frac{1}{f} \cos (\frac{A}{2}) = \frac{b + c}{2 b c}$

$\Rightarrow \frac{1}{f} \cos (\frac{A}{2}) = \frac{1}{2} (\frac{1}{b} + \frac{1}{c}) . . . (1)$

Similarly, we get

$\frac{1}{g} \cos \frac{B}{2} = \frac{1}{2} (\frac{1}{c} + \frac{1}{a}) . . . . (2)$

$\frac{1}{h} \cos \frac{C}{2} = \frac{1}{2} (\frac{1}{a} + \frac{1}{b}) . . . (3)$

Adding equations $(1), (2)$ and $(3)$ we get

$\frac{1}{f} \cos (\frac{A}{2}) + \frac{1}{g} \cos (\frac{B}{2}) + \frac{1}{h} \cos (\frac{C}{2})$

$= \frac{1}{2} [\frac{1}{b} + \frac{1}{c} + \frac{1}{c} + \frac{1}{a} + \frac{1}{a} + \frac{1}{b}]$

$= \frac{1}{2} [\frac{2}{a} + \frac{2}{b} + \frac{2}{c}]$

$= \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$

If $f, g, h$ are internal bisectors of the angles of a triangle $A B C$ , show that
$\frac{1}{f} \cos \frac{A}{2} + \frac{1}{g} \cos \frac{B}{2} + \frac{1}{h} \cos \frac{C}{2} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ .

Important Questions on Properties of Triangles

Learn and Prepare for any exam you want !

If f, g, h are internal bisectors of the angles of a triangle ABC, show that 1fcosA2+1gcosB2+1hcosC2=1a+1b+1c.

Important Questions on Properties of Triangles

Learn and Prepare for any exam you want !

If $f, g, h$ are internal bisectors of the angles of a triangle $A B C$ , show that
$\frac{1}{f} \cos \frac{A}{2} + \frac{1}{g} \cos \frac{B}{2} + \frac{1}{h} \cos \frac{C}{2} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ .