Question

In a right angled triangle $\Delta$ ABC with C as a right angle, a perpendicular CD is drawn to AB. The radii of the circles inscribed into the triangles ACD and BCD are equal to x and y respectively. Find the radius of circle inscribed into the $\Delta$ ABC.

Answer 1

Let x,y and r be radii of the circles inscribed into the ${\Delta }^2$ ACD, BCD ABC and respectively. Then from similar ${\Delta }^2$ ABC and ACD, we get
$\frac{r}{x}=\frac{AB}{AC}=\frac{c}{b},$ where b = $\frac{cx}{r}.$
Similarly from similar
$\Delta$ ABC and BCD, we get
$\frac{r}{y}=\frac{AB}{BC}=\frac{c}{a},$ where a = $\frac{cy}{r}$
Now $c^2=A{B}^2=a^2+b^2=\frac{c^2{y}^2}{r^2}+\frac{c^2{x}^2}{r^2}=\frac{c^2(x^2+y^2)}{r^2}$
This gives r = $\sqrt{x^2+y^2}$