Where Do They Meet?

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.