There is standart well known geometric way to add two points $P$ and $Q$ of elliptic curve, that is by drawing a straight through the two points $P$ and $Q$ and getting a third point $R$. Take symmetry of $R$ along the x-axis and we call $P+Q$ is our new point.
My question is why do we need to take symmetry? Why is not just $P+Q =R$?
My idea was if $P+Q$ is $R$ then look $2P$ i.e adding point with itself. So, draw tangent at point $P$ then intersection point between curve and line is again $P$ which implies $P=2P$ then $P = \mathcal O$ i.e point at infinity. This leads that we have only one point which is point at infinity.
Do you think there is a problem with my thought and answer?