Lesson Prove that if a, b, and c are the sides of a triangle, then so are sqrt(a), sqrt(b) and sqrt(c)

Prove that if a, b, and c are the sides of a triangle, then so are , , and

Problem 1

Let assume that  

     +  <= .    (1)


where "a", "b" and "c" are the sides of a triangle.


I want to lead it to CONTRADICTION.


Indeed, square both sides of (1). You will get


    a +  + b <= c,   or


    a + b <= c - .


Then even more so

    a + b < c.


But it contradicts to the triangle inequality  a + b > c.


Thus we proved that  a + b > c   IMPLIES    +  > .


It works for any combinations of sides of a triangle.


Thus we proved that the values  ,  ,   satisfy all triangle inequalities,

if  "a",  "b"  and  "c"  are the sides of a triangle.


It implies that if  "a",  "b"  and  "c"  are the sides of a triangle, then  ,  ,  and    form a triangle, too.

Problem 2

For n= 2 it is just proved in Problem 1 above.

So I will assume here that n >= 3.


Let assume that  

     +  <= .    (1)


where "a", "b" and "c" are the sides of a triangle.


I want to lead it to CONTRADICTION.


Indeed, raise both sides of (1) to degree n. You will get


    a +  + b <= c,   or


    a + b <= c - .


Then even more so

    a + b < c.


But it contradicts to the triangle inequality  a + b > c.


Thus we proved that  a + b > c   IMPLIES    +  > .


It works for any combinations of sides of a triangle.


Thus we proved that the values  ,    and    satisfy all triangle inequalities,

if  "a",  "b"  and  "c"  are the sides of a triangle.


It implies that if  "a",  "b"  and  "c"  are the sides of a triangle, then  ,    and    form a triangle, too.

Problem 3

Let  "a",  "b",  and  "c" are the sides of the triangle.


Then for any combination of sides, we have the "triangle inequality
Let assume that  

    a + b > c.    (1)


It implies

    f(a) + f(b) > f(a+b)   due to the property of the function f(x),  and then

                  f(a+b) > f(c)  due to monotonity of the function f(x).



Thus we proved that  a + b > c  implies  that  f(a) + f(b) > f(c).


So, we proved that if  "a",  "b"  and  "c"  are the sides of a triangle, then the values  f(a),  f(b)  and  f(c)  satisfy triangle inequalities.


It works for any combinations of sides of a triangle.



It implies that if  "a",  "b"  and  "c"  are the sides of a triangle, then  f(a),  f(b)  and  f(c)  form a triangle, too.