Will in a triangle ABC, the ex-circle w.r.t. A & the in-circle touches at the same point always? What in case of isosceles triangle? What is the condition for that? PLz xpln details.

Dear Padma Bhushan,

Solution:- It is not necessary that the ex-circle w.r.t. A and in-circle always touch at the same point to line BC. Even, in most of the cases, they do not touch at the same point.

In case of isosceles triangle (having AB = AC), ex-circle w.r.t. A and in-circle always touch at the same point to line BC. And that point is the mid point of BC. Note that, ex-circles w.r.t. B and C will not touch the in-circle at the same point to line AC and AB respectively (in this case).

In case of equilateral triangle (having AB = AC = BC), ex-circles w.r.t. A,B and C and in-circle always touch at the same point in line BC, AC and AB respectively.

 
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