IGO 2016 Elementary/3

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Thamim Zahin
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IGO 2016 Elementary/3

Unread post by Thamim Zahin » Tue Jan 10, 2017 2:15 pm

3. Suppose that $ABCD$ is a convex quadrilateral with no parallel sides. Make a parallelogram on each two consecutive sides. Show that among these $4$ new points, there is only one point inside the quadrilateral $ABCD$.
Last edited by Thamim Zahin on Wed Jan 11, 2017 8:08 pm, edited 1 time in total.
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Thamim Zahin
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Re: IGO 2016 Elementary/3

Unread post by Thamim Zahin » Wed Jan 11, 2017 7:57 pm

It is obvious that, if the angle of one side (the adjacent two angle of side) less than $180^o$, than the two parallelogram made by that side and the adjacent side are outside. Cause their angle sum is $180^o$. But the angle sum is less than $180^o$.

Now take one more side adjacent to the side we take earlier. So the sum of angle adjacent to that side is less than $180^o$. So, like before we will get one more parallelogram(we count one because we had already count one).

So, we have only one parallelogram left. The parallel side of that would definitely intersect inside. Because, the angle sum of the both side's adjacent angle is greater than $180^o$. And the sum of a parallelogram's one side's is $180^o$.

So, that means there will be only one point inside the quadrilateral.

$[Proved]$
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.

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