Question

The vertices of the triangles ABC are the points (6 , 0) ,( 0 , 6 ) and (7 , 7 ) prove that the equation of the circle inscribe in the triangle is
$x^2+y^2-$ 9x - 9y + 36 = 0

Answer 1

Centre is obtained by using the formula $(\frac{\sum a{x}_1}{\sum a},\frac{\sum a{y}_1}{\sum a})$ for incentre
$(\frac{72\sqrt{2}}{16\sqrt{2}},\frac{72\sqrt{2}}{16\sqrt{2}})$ or $(\frac{9}{2},\frac{9}{2})$
Radius is distance of centre from any side say AB whose equation by intercepts from is
$\frac{x}{y}+\frac{y}{x}$ = 1 or x + y - 6 = 0
$\therefore r=\frac{\frac{9}{2}+\frac{9}{2}-6}{\sqrt{2}}=\frac{3}{\sqrt{2}}$
$\therefore$ Circle is
${(x-\frac{9}{2})}^2+{(y-\frac{9}{2})}^2={\left(\frac{3}{\sqrt{2}}\right)}^2$
or $x^2+y^2$ - 9x - 9y + 36 = 0