Finding if two points on elliptic curve are related On a given elliptic curve I have some points that are defined like this:
Where
$$A=a*G$$
$$B=b*G$$
$$C=(a+b)*G$$
$$D=d*G$$
$$E=(a+d)*G$$
So finally I have two equations like below:
$$A+B=C$$
$$A+D=E$$

Given the values $(C,E,A)$, is there any way I can prove that point $A$ is common in both the points $C$ and $E$?

Or can I also prove that point $A$ is part of the $C$ and $E$?
No you can't.

I'll make the argument over the integers, it directly transfers to ECC points.

Suppose you have two integers $c,e\in\mathbb Z$. You can now prove for every $a\in\mathbb Z$ that $c$ and $e$ have $a$ "in common".

For this you simply pick $b=c-a$ and $d=e-a$ and now you have the "proof" that both have $a$ in common. Note how no restrictions whatsoever have been placed on $a,c,e$ so it works with all triples.