Find The Length

Geometry Level 5

In \(\triangle ABC,\) \(BC=9, CA=8, AB=6.\) The incircle of \(\triangle ABC\) touches lines \(AB,AC\) at points \(Z,Y\) respectively. Let \(R\) be the unique point such that \(BCYR\) is a parallelogram. Lines \(BY,CZ\) intersect at \(G.\) Given that \(GR= \sqrt{\dfrac{p}{q}}\) for some coprime positive integers \(p,q,\) find the last three digits of \(p+q.\)

Extra credit

Let \(S\) be the unique point such that \(BCZS\) is a parallelogram.

  • Show that \(GR=GS.\)
  • Show that \(RZSY\) is a parallelogram.

Details and assumptions

  • This problem is inspired by ISL 2009 G3.

  • You might use WolframAlpha for the calculations, this problem really gets tedious after a while.

  • The last digit of the answer is \(7.\) Hopefully that will help get rid of plugging / calculation errors. (I don't want people commenting again saying that they plugged in wrong values or something.)

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