This looks ugly (but I assure you it isn't)

I apologize for making the question somewhat deceiving: the equation is actually a very fancy way of saying $1$ ! However, if I asked you to find the constant value for it, it would be trivial to find it because you could just plug in some values and guess $1$ .

If we fix $a, b, c$ then the expression becomes a quadratic with a variable of $A$ . Note that at $A=a, A=b, A=c$ two of the large terms become zero because they contain $A-a$ , $A-b$ , or $A-c$ on top and the last term has a matching numerator and denominator. Thus, at these values the expression is $1$ .

Three points define a quadratic equation! The only one that fits is that for any $A$ the quadratic yields $1$ , therefore the expression simplifies to $1$ .