We know that a + b + c = 2

In any triangle, whichever side you choose, the length of that side must be less than the sum of the lengths of other two sides, so we can tell that

a + b > c, b+ c > a and c + a > b.

Since a + b = 2 – c, it must be true that

2 – c > c, and so c < 1.

The same method shows that a < 1 and b < 1 as well.

Therefore 1 – a > 0, 1 – b > 0 and 1 – c > 0, so the product (1 – a)(1 – b )(1 – c) must be positive.

Now we can just simplify:

(1 – a) (1 – b ) (1 – c) > 0

1 – a – b – c + ab + bc + ca – abc > 0 (recall that a + b + c = 2)

1 – 2 + ab + bc + ca > abc

Hence, tidying up,

ab + bc + ca > abc + 1.

Challenge 341: Triangle Teaser