This might be a very basic question, but I've failed in finding an explanation on the web ...

How is it possible that a variable (Var1) that significantly correlates with two different variables (Var2 and Var3), would not correlated also with a fourth variable (Var4) which is the result of dividing the later two variables (Var4 = Var2/Var3)?

For example:
I have an image in which one of the properties is "Quality".
I use this image to obtain two measurements: $M\_1$ and $M\_2$. From $M\_1$ and $M\_2$ I calculate $F\_1$ (which is my endpoint) by dividing $M\_1$ by $M\_2$ ($F\_1 = M\_1/M\_2$).

Although I get a significant correlation between "Quality" to both $M\_1$ and $M\_2$ (Spearman's -0.625, P<0.0001 and -0.636, P<0.0001 respectively), the correlation between $F\_1$ and "Quality" is non-significant (-0.095, p=0.565).
It should also be noted that $M\_1$ and $M\_2$ strongly correlate with each other (0.692, P<0.0001).