triangle inequality theorem states Sum of any $2$ sides must be greater than $3$rd side.
I am doing exercises in my text book. For example, I am checking if $10,3,5$ can form a triangle.
$10+3 > 5$
$10+5 > 3$
$5+3 > 10$ : False
Third condition is false and hence triangle cannot be formed.
For every problem, all the conditions are tested or we stop when any of the condition fails.
But, can't we just say "Sum of shorter sides must be greater than the longest side" and always check this condition only?
i.e., we check add $5$ and $3$ (shorter sides) and compare with 10(longer side). if this test pass, other two tests must also pass, right?
In summary, I need to add two shorter sides and compare with the longer side instead of testing all three conditions. Am I correct?
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$\begingroup$Yes, that's true.
Let $ a <= b <= c $, if $ c < a + b $ then:
$$ b <= c < a + b <= a + c $$ Therefore $$ b < a + c $$
Likewise: $$ a <= c < a + b <= c + b $$ $$ a < c + b $$
To us humans, it's easy enough to immediately find the biggest of $3$ numbers, and easier to write just one comparison than $3$. However, if you think about it, to actually find who's the biggest one you have to do $2$ comparisons already, however trivial they may be. So it's not THAT big of an improvement, at least from a machine's POV and if you can tolerate the extra writing.
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