If $$abc0 $$ such taht
$$p_1 = \sqrt{(a+b)^2+c^2}$$,
$$p_2= \sqrt{(a+c)^2+b^2}$$,
$$ p_3 =\sqrt{(b+c)^2+a^2},$$
Compare the size of $$p_1$$, $$p_2$$ and $$p_3$$

Question

If $$a>b>c>0 $$ such taht
$$p_1 = \sqrt{(a+b)^2+c^2}$$,  
$$p_2= \sqrt{(a+c)^2+b^2}$$, 
$$ p_3 =\sqrt{(b+c)^2+a^2},$$ 
Compare the size of $$p_1$$, $$p_2$$ and $$p_3$$

Uniteasy Admin · Accepted Answer

$$\sqrt{(a+b)^2+c^2}$$
$$=\sqrt{a^2+b^2+c^2+2ab} $$
$$=\sqrt{a^2+b^2+c^2+2abc\cdot\dfrac{1}{c}} $$n
$$\sqrt{(a+c)^2+b^2} $$
$$=\sqrt{a^2+b^2+c^2+2abc\cdot\dfrac{1}{b}} $$n
$$\sqrt{(b+c)^2+a^2}$$
$$ =\sqrt{a^2+b^2+c^2+2abc\cdot\dfrac{1}{a}} $$n
$$	herefore p_1>p_2>p_3 $$

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