The area #\Delta# of triangle with vertices #(x_1, y_1)\equiv(4, 5)#, #(x_2, y_2)\equiv(1, 2)# & #(x_3, y_3)\equiv(5, 3)# is given by following formula
#\Delta=1/2|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|#
#=1/2|4(2-3)+1(3-5)+5(5-2)|#
#=4.5#
Now, the lengths of all three sides say #a, b# & #c# of given triangle are computed by using distance formula as follows
#a=\sqrt{(4-1)^2+(5-2)^2}=3\sqrt2#
#b=\sqrt{(4-5)^2+(5-3)^2}=\sqrt5#
#c=\sqrt{(1-5)^2+(2-3)^2}=\sqrt17#
hence, the semi-perimeter #s# of given triangle is computed as follows
#s=\frac{a+b+c}{2}#
#=\frac{3\sqrt2+\sqrt5+\sqrt17}{2}=5.3#
hence, the radius of inscribed circle is given as
#\frac{\Delta}{s}#
#=\frac{4.5}{5.3}#
#=0.849#