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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