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Just an informal question out of curiosity.

Let's consider the simplest case of a circle, there is an external point P, and consider one of the 2 tangents possible through point P.

Does it even matter if tangent is a line, line-segment or ray? I mean, with tangent we are quite flexible, we treat them as it deems necessary. But, is there a fixed definition of tangent that attributes it to the one of 3 possibilities?

Edit : I discussed this question with my friends and a few teachers. Here is the statistic - Most of them seem to agree on tangent being a 'ray'. 'Line' is also a good choice. But almost nobody seems to go with 'line segment'. Personally, I prefer line segment because it gives us 2 informations - first, the origination point P and second, the end point (which lies on curve).

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  • $\begingroup$ From what I can see from Wiki, "tangent" is like an adjective. We have tangent line, tangent planes, etc. I think in your simple case, you can come up with two tangent lines, tangent line-segments or tangent rays, depending on how you draw the lines. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Mar 11 '17 at 8:58
  • $\begingroup$ I agree. But the question is if the tangent is 'pre-defined' to be a line, ray or line-segment or not. Your statement seems to say it's not pre-defined. I agree, this might very well be the possibility but the problem is we aren't sure. $\endgroup$ – Sarthak123 Mar 11 '17 at 9:05
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    $\begingroup$ In my experience "tangent" is used both as an adjective and as a noun. When used as a noun, I've never really though about whether it was the complete line, just a segment of it or ..., I think I would say it's the line. $\endgroup$ – Henrik Mar 11 '17 at 9:09
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Regarding your informal question, I think we don't have a formal classification on whether a tangent is a line, or a ray, or a line segment. All we can do is to roughly guess the type of that “line” by considering its associated description.

enter image description here

The red line, $\alpha$, is a tangent to the circle and touches it at A. That tangent is a line.

The blue tangent pair, PR and PS, is the tangents to the circle from the external point P. It should be regarded as rays generated from P (provided the dotted parts have no significance).

We also interpret The length of the tangent QT is the length of a line- segment. Of course, PR and PS are also line-segments.

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