Question:
In acute $\Delta ABC$, let $D$ be the foot of the altitude from $A$ to $BC$, and let $\overline{AD}$ intersect the circumcircle of $\Delta ABC$ at $E$.
Let the circle with diameter $AE$ intersect lines $AB$ and $AC$ at $N$ and $M$, respectively. Given that $DB=3NB$ and $MA=5NA$,
then the value of $\displaystyle \frac{DC}{MC} $ can be written in simplest form as $\displaystyle \frac{a}{b}$. What is the value of $a-b$?
My attempt to draw: Please guide me.
On
The problem statement says, "Let the circle with diameter $AE$ intersect lines $AB$ and $AC$" (emphasis added by me).
If the statement had said "sides $AB$ and $AC$" then your constructions would be counterexamples. But the line $AC$ is generally considered to be the line through $A$ and $C$ extended indefinitely in any direction. Since the circle with diameter $AE$ intersects that circle at $A$ and is not tangent to the line at $A$, it will certainly intersect the line at one other point. That point may not be between $A$ and $C$ but it will still exist.
It does indeed appear to indicate that there is a typo in the question as posed. The circle with the diameter $AE$ does not have to intersect $AB$ and $AC$ at all, and so in particular the point $M$ does not exist.