Properties of triangles and circles!

This probably looks like a lot of work, but honestly, you could do the whole thing in your head...

Let's start by finding $y$ . The first thing I notice is that I need to find the measure of $\angle CAD$ , because $AE$ bisects the angle, so to find $m\angle EAD$ , i'd just divide by two. What I notice immediately is that $\widehat CD$ subtends both $\angle CAD$ AND $\angle CBA$ . Lucky for me, there's a theorem for that: the central angle subtended by two points on a circle is twice the inscribed angle subtended by those points. The inscribed angle is $\angle CBA$ , which is of course equal to $37 ^ \circ$ . $37*2=m\angle CAD$ . But hey, later on I am going to divide that same angle by 2 to get $m\angle EAD$ ! So now, if we divide on both sides, we have $37=\frac{m\angle CAD}{2}$ , and we can use the substitution property (since $m\angle EAD=\frac{m\angle CAD}{2}$ ) to get $m\angle EAD=37 ^ \circ$ . $\triangle CAD$ is isosceles since it is created by two radii. This means that the base angles are congruent. With the Triangle Sum Theorem, we can set this up as $180-74=m\angle ACD*2$ . Solving this, we get $m\angle ACD=53 ^ \circ$ . Since $y=m\angle EAD+m\angle ACD$ , we can substitute $y=37+53$ , and we get $y=90$ .

Now, to find $z$ , which is the length of $AE$ . $CE=ED$ , since it is bisected by $AE$ (There's a theorem about radii perpendicular to chords, but I decided to make it a perpendicular bisector anyways). So $ED=900$ . $AD=975$ because it is a radius, like $AB=975$ . $AE$ is perpendicular to $ED$ , and we see that this creates a right triangle! But how can we solve for such big numbers? The first thing that we always look for is a Pythagorean triple, and we look for proof that it is just one of the first few triples multiplied by something! I try dividing the hypotenuse by 5, and $ED$ by 4, then 3. Do any of the results match up? Nope! I get 195, 225, and 300. So, I will move on to the next triple, 5-12-13. Dividing 975 by 13, I get 75. Dividing 900 by 12, I also get 75. This coincidence shows us that $AE=5*75=375=z$ . So, $z=375$ .

Now we have $z=375$ and $y=90$ . We can put these together in the fraction $\frac{375}{90}$ , which, in simplest form, would be $\frac{25}{6}$ .

So, $a=25$ and $b=6$ . All we need to do is subtract $25-6=\boxed{19}$ .

So $\boxed{19}$ is the answer!