Parallel lines through inner point of triangle - Austria '10

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Parallel lines through inner point of triangle - Austria '10

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#1 h May 13, 2010, 5:45 PM • 2 Y

Y by Adventure10, Mango247

The the parallel lines through an inner point $P$ of triangle $\triangle ABC$ split the triangle into three parallelograms and three triangles adjacent to the sides of $\triangle ABC$ .
(a) Show that if $P$ is the incenter, the perimeter of each of the three small triangles equals the length of the adjacent side.
(b) For a given triangle $\triangle ABC$ , determine all inner points $P$ such that the perimeter of each of the three small triangles equals the length of the adjacent side.
(c) For which inner point does the sum of the areas of the three small triangles attain a minimum?

(41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 4)

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#2 h May 13, 2010, 7:39 PM • 1 Y

Y by Adventure10

Outline solution.

a) Call I the incenter of $ABC$ and parallels from I to AC,AB cut $BC$ in $M,N$ . Easy to see that $IMC$ and $IMB$ are isosceles $\Rightarrow$ $IM=MC$ , $IN=NB$ , We have then $IM+IN+NM=MC+BN+NM=BC$ . Similarly we're done.

b) I think that the only point that satisfies this, is the incenter of $\Delta ABC$ (not sure about it)

c) Use RMS-AM to get that P has to be the center of gravity of $\Delta ABC$

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