Is there a standard notation for the plane passing through three non-collinear points?

Question

Is there a "standard" notation for designating the unique plane passing through given three non-collinear points $A$, $B$, and $C$? Lacking a standard notation, are there notations used for this purpose in some popular maths textbooks/papers or in some textbooks/papers written by distinguished mathematicians?

For comparison, $\overleftrightarrow{AB}$ is a standard notation for the unique line passing through the distinct points $A$ and $B$.

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I'll repeat what I wrote in a comment below: I am interested in a notation that is viable in a Synthetic Geometry framework, where there's no coordinate system. Of course, there's nothing in my original question to indicate there's a coordinate system that can be used, so this remark is just for emphasis.

Answer 1

No; the closest I've seen is "Plane defined by $ABC$" or "Plane defined by $\mathbf{A}\mathbf{B}\mathbf{C}$" or "Plane defined by points $\vec{A}$, $\vec{B}$, and $\vec{C}$".

Furthermore, I claim that it is better to use a simple, consistent notation, and describe it at the first use, than try to relying on "commonly recognized notation". This is because in various subfields from pure linear algebra to physics, very different notation is "standard" anyway; but in all subfields, the best articles always describe the notation (and abbreviations) at first use.

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OP is not interested in any suggestions, but personally, I would use something like

Plane $ABC$ is the plane passing through three non-collinear points $A$, $B$, and $C$; the locus of points $P$ such that $P \cdot\left(\overline{AB}\times\overline{AC}\right) - A \cdot \left(\overline{AB}\times\overline{AC}\right) = 0$.

for the first instance, and then just "plane $ABC$" afterwards, assuming notation $\overline{AB} = B - A$ is acceptable; otherwise,

Plane $\mathbf{A}\mathbf{B}\mathbf{C}$ is the plane passing through three non-collinear points $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$; that is, the locus of points $\mathbf{P}$ such that $\mathbf{P} \cdot \bigl(\left(\mathbf{B}-\mathbf{A}\right) \times \left(\mathbf{C}-\mathbf{A}\right)\bigr) - \mathbf{A} \cdot \bigl(\left(\mathbf{B}-\mathbf{A}\right) \times \left(\mathbf{C}-\mathbf{A}\right)\bigr) = 0$.

Note that the above is equivalent to $\left(\mathbf{P} - \mathbf{A}\right) \cdot \left(\mathbf{B} - \mathbf{A}\right) \times \left(\mathbf{C} - \mathbf{A}\right) = 0$, a triple product, but this form is not as recognizable as the standard implicit definition of a plane in vector notation. Thus, I would advise against using this form even if it is more elegant.

For those who are not aware of it, the plane normal $\mathbf{N}$ is obtained by the cross product of any pair of edges between the three points, and the signed distance $d$ (in units of the plane normal length) is the dot product between the plane normal and any of the three points (all three yield the same distance $d$). These are then used in the implicit equation of the plane: the locus of points $\mathbf{P}$ where $\mathbf{P} \cdot \mathbf{N} - d = 0$.