a, b and c are the sides of a right triangle, where c is the hypotenus

1.	a, b and c are the sides of a right triangle, where c is the hypotenuse. A circle, of radius r, touches thea+b-csides of the triangle. Prove thatr=\frac{a+b-c}{2}
Answer» Let the circle touches the sidesAB,BC andCA oftriangle ABCat D, E and F Since lengths of tangents drawn from an external point are equal We have AD=AF, BD=BE and CE=CF Similarly EB=BD=r Then we have c=AF+FC ⇒ c=AD+CE ⇒ c= (AB-DB)(CB-EB) ⇒c = a-r +b-r ⇒ 2r=a+b-c r =( a+b-c)/2

a, b and c are the sides of a right triangle, where c is the hypotenuse. A circle, of radius r, touches thea+b-csides of the triangle. Prove thatr=\frac{a+b-c}{2}

Answer»

Let the circle touches the sidesAB,BC andCA oftriangle ABCat D, E and F

Since lengths of tangents drawn from an external point are equal We have

AD=AF, BD=BE and CE=CF

Similarly EB=BD=r

Then we have c=AF+FC

⇒ c=AD+CE

⇒ c= (AB-DB)(CB-EB)

⇒c = a-r +b-r

⇒ 2r=a+b-c

r =( a+b-c)/2