Right triangle inradius and circumradius

 In a right triangle if inradius  = r, hypotenuse  = c and legs are a,b then:
 r = (a + b - c)/2

Proof:



From the center of the incircle, the perpendiculars are dropped on the 2 legs (a,b).
It creates a square and splits the base in 2 parts r, a-r and similarly the other leg is split into r,b-r.
Since 2 tangents from one point are equal => c = a - r + b - r =>
r = (a + b - c)/2

Circumradius of a right triangle = c/2 where c is hypotenuse.
Proof:


Easy by co-ordinate geometry.
A = (0,0) B = (a,0) C = (0,b)
Midpoint of BC = a/2,b/2
OA = OB = OC = sqrt(a^2/4 + b^2/4)
=> O is the circumcenter and OA = Hypotenuse/2  = Circumradius.

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