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Per an educational source, to quote:

Tangent, in geometry, straight line (or smooth curve) that touches a given curve at one point; at that point the slope of the curve is equal to that of the tangent. A tangent line may be considered the limiting position of a secant line as the two points at which it crosses the curve approach one another.

Note, the definition implies the ostensible equivalent of the tangent to a tangent line. The latter apparently is explicitly defined as a limit of a secant line, which is comprised of two points.

This inherently makes sense as two points determine a line.

So, the concept of a 'tangent' appears to be short for a tangent line, which by a limiting definition, is explicitly defined.

So, my answer is generally no, as geometry is usually taught first.

My argument is that the concept of a 'tangent' (implying a tangent line) at a point (which does not determine a line) is not clear for those not acquainted calculus-based concept of a limit. In the current context, the definition explicitly cites to quote again "limiting position of a secant line as the two points at which it crosses the curve approach one another".

I also cite a related prior question "how to draw a tangent line to a curve fit" which, by the very question, supports my answer. Note: the provided answer makes use of the explicit knowledge of the equation for the curve in question [EDIT] which is not always the case. [EDIT][EDIT] To expound on this comment, per Wikipedia on tangent, to quote:

The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century. In the second book of his Geometry, René Descartes[8] said of the problem of constructing the tangent to a curve, "And I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know".[9]

AJKOER
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    tangent line to a curve can touch it at multiple points, for instance the tangent line to a straight line. – Jamāl Dec 19 '20 at 17:18
  • Thanks Jamal. For the sake of the question, I restricted my reference frame to the cited educational source. Note per Wikipedia to quote: "Leibniz defined it as the line through a pair of infinitely close points on the curve.[1]" at https://en.wikipedia.org/wiki/Tangent, which is in agreement with my cited source. – AJKOER Dec 19 '20 at 17:20
  • This belongs on https://matheducators.stackexchange.com/ Voting to close. – saulspatz Dec 19 '20 at 17:24
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    To make matters worse (when trying to deduce something precise from statements not intended to be precise, or made by someone not sufficiently knowledgeable), there is the issue of what "limiting position of a secant line as the two points at which it crosses the curve approach one another" means. In older analytic geometry literature it meant what is now called the strong derivative (see also the links I cite in my comment to this question), which is a stronger notion of being differentiable. – Dave L. Renfro Dec 19 '20 at 17:29
  • @Dave L. Renfro Thanks for the mathematical extension. It is that depth and my sense of vagueness on the definition that I felt that this was topic belongs in this forum. – AJKOER Dec 19 '20 at 17:42
  • For the things done in the "related prior question" you cited, it's not really an issue, being somewhat akin to worrying mathematically about whether the interface between air and water is air, or water, or some mixture of each. FYI, mathematically the issue with the ordinary vs. strong derivative is whether, in taking a limit of secant lines, we always have the limiting point as one of the two points used to define the secant line, or can the two points used to define the secant line be any two points that get arbitrarily close to the limiting point. (continued) – Dave L. Renfro Dec 19 '20 at 17:54
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    See pp. 104-105 of Fowler's 1920 book The Elementary Differential Geometry of Plane Curves. This book, by the way, was probably the first non-research level publication (paper or book) that brought attention to this issue, which was discovered by the Italian mathematician Peano sometime around or before 1892. – Dave L. Renfro Dec 19 '20 at 17:54
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    "the concept of a 'tangent' (implying a tangent line) at a point (which does not determine a line) is not clear for those not acquainted calculus-based concept of a limit." Yes, that's correct, you need calculus to define the tangent line to the graph of a function at a given point. The tangent line to the graph of $f$ at the point $(a,f(a))$ is by definition the line through this point with slope $f'(a)$. – littleO Dec 19 '20 at 18:17
  • LittleO: Yes, I agree with your comment which is, however, qualified per my edited comment (the very last words in my question), assuming an available specification of the underlying function generating the curve itself and also the existence of the derivative at point 'a'. – AJKOER Dec 19 '20 at 18:54
  • Further: Per Fowler's book, Note A, Page 104, a clear reference to the definition of Strong Derivative as referenced above. – AJKOER Dec 19 '20 at 19:03
  • I have further commented on the non-functional based approach to a tangent line construction in a further edit to my question, for those interested. – AJKOER Dec 19 '20 at 19:20

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COMMENT

►From viewpoint of tangent as a right line touching a given curve at one point, let me give two examples of curves. $$(1)\hspace2cmy= |x-3|+2\\(2)\hspace2cmy^2=(x-3)^3$$ The curve $(1)$ has in the point $(3,2)$ infinitely many tangents and so is for the curve $(2)$ in the point $(3,0)$.

►From viewpoint of tangent as limiting position of secants we have for the curve $(1)$ only two (distinct) tangents while for the curve $(2)$ there are just one tangent. (nevertheless the curve is singular and is not an elliptic curve).

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►Is for many reasons that the definition of tangent in a point of a curve is given for differentiable curves in the considered point.

Piquito
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