# Where Do They Meet?

Geometry Level 5

Let $$\triangle ABC$$ be an acute triangle with side lengths $$BC= 6, CA= 7, AB= 8.$$ Let $$B', C'$$ be the diametrically opposite points of $$B,C$$ respectively on the circumcircle of $$\triangle ABC.$$ Line $$B'C'$$ intersects lines $$CA$$ and $$AB$$ at points $$D,E$$ respectively. Let $$D',E'$$ be the feet of perpendiculars from $$D, E$$ respectively on $$BC.$$ The line passing through $$E'$$ parallel to $$AB$$ and the line passing through $$D'$$ parallel to $$CA$$ meet at $$X.$$ If $$AX^2$$ is equal to $$\dfrac{a}{b}$$ for some coprime positive integers $$a,b,$$ find $$a+b.$$

Details and assumptions

• The diametrically opposite point of a point $$X$$ lying on a circle $$\omega$$ is the unique point $$X'$$ apart from $$X$$ such that $$XX'$$ is a diameter of $$\omega.$$

• The diagram shown is not accurate.

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