Given that a, b, c are the length of three sides of triangle ABC, try to simplify: the absolute value of a-b-c plus the absolute value of a+c-b

Given that a, b, c are the length of three sides of triangle ABC, try to simplify: the absolute value of a-b-c plus the absolute value of a+c-b

Because the triangle 2 side difference is less than the third side
Triangle 2 sides and greater than 3rd side
So a-c < b < a+c
|A-b-c c-b|=|(a-c)-b (a+c)-b|
=B-a+c+a+c-b
=2C

Because the triangle 2 side difference is less than the third side
Triangle 2 sides and greater than 3rd side
So a-c < b < a+c
|A-b-c c-b|=|(a-c)-b (a+c)-b|
= B-a+c+a+c-b
=2C

If abc is the trilateral length of triangular abc, the absolute value of a minus b minus c plus the absolute value of b minus c minus a plus the absolute value of c minus a minus b is reduced. If abc is the triangular length of abc, the absolute value of a minus b minus c plus the absolute value of b minus c minus a plus the absolute value of c minus a minus b is reduced.

|A-b-c||b-c-a||c-a-b|
=-(A-b-c)-(b-c-a)-(c-a-b)
=-A+b+c-b+c+a-c+a+b
=A+b+c (the difference between any two sides of the triangle is less than the third side)