Question

If a,b,c are the sides of a right triangle , where c is the hypotenuse. Prove that the radius r of the circle which touches the sides of the triangle is given by:`r=(a+b-c)/2`

Answer 1

Since `OE_|_AC` and `OD_|_BC`
(radius through point of contact is `_|_ `to the tangent)
`:." "squareODCE` is a square of side r.
`:." "BD = a-r` and `AE = b-r`
Now, BC, AC and AB are the tangents of a circle.
`:." "BC=BD`
(length of tangents from an external point are equal)
`=a-r`
and `AC=AE" "`(length of rangents from an external point are equal)
`= b-r`
Now `C = AB=BC+AC`
`implies" "C=a-r+b-r" "implies" "2r=a+b-c" "implies" "r=(a+b-c)/(2)`
Hence Proved.