# 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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