Triangle contained in another triangle, sharing one angle with constraints on sides

Question

I'm trying to solve another triangle problem:

Knowing all the sides and angles of a triangle $ABC$ i'm trying to build a triangle $AB'C'$ such as

It shares the angle $A$

$a' = 2x$

$b' = b - x$

$c' = c - x$

but I can't figure out how to solve $x$

any idea ?

thanks

Answer 1

Use cosine law

$$2(b-x)(c-x)\cos A=(b-x)^2+(c-x)^2-(2x)^2$$

Solve this quadratic equation and you get $x$.