I use Mathematica to add two points graphically on the elliptic curve $y^2 = x^3 + 3x + 8$ over $\mathbb{F}_{13}$. Specifically, I'd like to illustrate $(1,8)+(2,10)=(1,5)=(1,-8)$, but on first glance, $(2,3)=(2,-10)$ could also be a solution, because there are no other points of intersection on the line through $(1,8)$ and $(2,10)$.
$(1,5)$ is the correct solution because the equation to solve the addition algebraically, $$ (8+2(x-1))^2 = x^3 + 3x + 8 $$ factors $$ 12(x+11)(x+12)^2 $$ so $x=1$ is the third root (in $\mathbb{F}_{13}$).
Question: How can I recognize this double root if I interpret the addition geometrically by drawing a line through the points to be added in $\mathbb{F}_{13}^2$?
As $(1,8)=(1,-5)$ and $(2,10)=(2,-3)$ I used reuns' idea, switched to the curve $$y^2=x^3-23x+47,$$ (observe that the coefficients are congruent to those of the original), and produced the following image. The red line is connecting $(1,-5)$ and $(2,-3)$, both now on the curve.
You see that due to the relative proximity of the points $(1,-5)$ and $(2,-3)$ on the red line, it is a bit taxing to "see" that this line is actually a tangent to the elliptic curve at the first point. Zooming in does reveal that at the second point there is no tangency.
If your audience has been exposed to the implicit function theorem, you can explain how this (real) variant of the curve has a tangent of slope $$ \frac{dy}{dx}=\frac{3x_0^2-23}{2y_0}=\frac{11}6 $$ at the point $(x_0,y_0)=(2,-3)$. Very close to the slope $m=2$ of the red line, explaining the difficulty.
Unfortunately this visualization is probably not as clear as you might want.