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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?

Fuat Ray
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  • Associativity fails; say a line interesects at three distinct points, $P$, $Q$, and $R$. Then $(P+Q)+R$ would be the third point of intersection of the tangent through $R$, and $P+(Q+R)$ would be third point of intersection of the tangent through $P$. Those are distinct (since a line can only intersect the elliptic curve at three points, counting multiplicity), so you will generally get different answers from $(P+Q)+R$ and $P+(Q+R)$. – Arturo Magidin Feb 12 '24 at 20:10

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