Incircle And Excircle Of a Triangle

Incircle And Excircle Of a Triangle

Some other formulae which the relation among \( r,\,\,r_1 ,\,\,\,r_2 \,\,\& r_3 \)
1. i)\( \frac{1} {r} = \frac{1} {{r_1 }} + \frac{1} {{r_2 }} + \frac{1} {{r_3 }} \)            ii)\( r.r_1 .r_2 .r_3 = \Delta ^2 \)               iii)\( r_1 r_2 + r_2 r_3 + r_3 r_1 = s^2 \)               iv)\( r(r_1 + r_2 + r_3 ) = ab + bc + ca - s^2 \)
v)\( (r_1 - r)(r_2 + r_3 ) = a^2 \)
\( (r_2 - r)(r_3 + r_1 ) = b^2 \)
\( (r_3 - r)(r_1 + r_2 ) = c^2 \)

vi)\( a = (r_2 + r_3 )\sqrt {\frac{{rr_1 }} {{r_2 r_3 }}} \)
\( b = (r_3 + r_1 )\sqrt {\frac{{rr_2 }} {{r_3 r_1 }}} \) 
\( c = (r_1 + r_2 )\sqrt {\frac{{rr_3 }} {{r_1 r_2 }}} \)

vii)   
viii)\( r_1 + r_2 + r_3 - r = 4R \) \( r + r_2 + r_3 - r_1 = 4R\cos A \)
\( r + r_3 + r_1 - r_2 = 4R\cos B \) \( r + r_1 + r_2 - r_3 = 4R\cos C \)

2. In an equilateral triangle of side a
(i) \( Area = \frac{{\sqrt 3 }} {4}a^2 \)       (ii) R=a/\( \sqrt(3)\)   (iii) r=R/2    (iv)\( r_1 = r_2 = r_3 = \frac{3} {2}R \)   (v)\( r:R:r_1 = 1:2:3 \)

3.    (i)  The distance between circum centre ‘s’ and orthocentre ‘O’ is
i.e.,\( OS = R\sqrt {1 - 8\cos A\cos B\cos C} \)
ii) Distance between circum centre and In centre

iii) Distance between circum centre  and excentres
\( I_1 S = \sqrt {R^2 + 2r_1 R} \)    \( I_2 S = \sqrt {R^2 + 2r_2 R} \)    \( I_3 S = \sqrt {R^2 + 2r_3 R} \)

4.     If AD is the median of \( \Delta ABC \) then \( AB^2 + AC^2 = 2(AD^2 + DC^2 ) \)
Length of median through \( A = \frac{1} {2}\sqrt {2b^2 + 2c^2 - a^2 } \)            
\( = \frac{1} {2}\sqrt {b^2 + c^2 + 2bc\cos A} \)
Length of median through \( B = \frac{1} {2}\sqrt {c^2 + a^2 + 2ac\cos B} \)
\( = \frac{1} {2}\sqrt {2c^2 + 2a^2 - b^2 } \)
Length of median through \( C = \frac{1} {2}\sqrt {2a^2 + 2b^2 - c^2 } \) \( = \frac{1} {2}\sqrt {a^2 + b^2 + 2ab\cos C} \)

Some Examples :
1. In \( \Delta ABC \), if a = 30 , b = 24, c = 18 then the ratio \( \frac{1} {{r_1 }}:\frac{1} {{r_2 }}:\frac{1} {{r_3 }} \)
Solution :
\( s = \frac{{a + b + c}} {2} = \frac{{30 + 24 + 18}} {2} = \frac{{72}} {2} = 36 \)
\( r_1 :r_2 :r_3 = \frac{\Delta } {{s - a}}:\frac{\Delta } {{s - b}}:\frac{\Delta } {{s - c}} \)
\( = \frac{1} {{s - a}}:\frac{1} {{s - b}}:\frac{1} {{s - c}} \)
\( = \frac{1} {6}:\frac{1} {{12}}:\frac{1} {{18}} \)
\( \frac{1} {{r_1 }}:\frac{1} {{r_2 }}:\frac{1} {{r_3 }} = 1:2:3 \)

2.    S.T \( r(r_1 + r_2 + r_3 ) = ab + bc + ca - s^2 \)
\( r(r_1 + r_2 + r_3 ) = \frac{\Delta } {s}\left( {\frac{\Delta } {{s - a}} + \frac{\Delta } {{s - b}} + \frac{\Delta } {{s - c}}} \right) \)
\( = \frac{\Delta } {s}.\Delta \left( {\frac{1} {{s - a}} + \frac{1} {{s - b}} + \frac{1} {{s - c}}} \right) \)
\( = \frac{{\Delta ^2 }} {s}\left( {\frac{{(s - b)(s - c) + (s - a)(s - c) + (s - a)(s - b)}} {{(s - a)(s - b)(s - c)}}} \right) \)
\( = \frac{{\Delta ^2 }} {s}\left( {\frac{{3s^2 - 2s(a + b + c) + ab + bc + ca}} {{(s - a)(s - b)(s - c)}}} \right) \)

\( = \frac{{\Delta ^2 }} {{\Delta ^2 }}(3s^2 - 2s(a + b + c) + ab + bc + ca) \) =(3s²-2s(2s)+ab+bc+ca)
\( r(r_1 + r_2 + r_3 ) = ab + bc + ca - s^2 \)