Question

Given a right triangle with ∠A = 90°. Let M be the mid-point of BC. If the in radii of the triangle ABM and ACM are  r₁ and r₂ then find the range of r₁/r₂.

Answer 1

Area (∆AMC) = Area(∆AMB)

(BC)/2 = AM = MB = a/2

AC = 2(AE) = b, AD = DB = a/2

In radii of ∆AMB =  ∆/s

= 1/2((Area(∆ABC))/((a + c)/2)) 

= Area(∆ABC)/(a + c)

= bc/(a + c)

In radius of ∆AMC = 1/2((Area(∆ABC))/(a + c/2))

Area(∆ABC)/(a + c) = bc/(a + b)