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(a) If a function $g$ is continuous on the closed interval $[u,v]$, where $u<v$, and differentiable on the open interval $(u,v)$, then there exists a point $c$ in $(u,v)$ such that

$$g(v)=g(u)+g′(c)(v-u)$$

The expression $(g(v)-g(u))/(v-u)$ gives the slope of the line joining the points $(u,g(u))$ and $(v,g(v))$, which is a chord of the graph of $g$, while $g′(c)$ gives the slope of the tangent to the curve at the point $(c,g(c))$. Thus the Mean value theorem says that given any chord of a smooth curve, we can find a point lying between the end-points of the curve such that the tangent at that point is parallel to the chord.

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One area I know of where you can use chords and tangents to geometrically interpret addition in a group structure is with elliptic curves.

The way we define point addition on an elliptic curve is by taking a secant (a secant is just a continuation of a chord beyond its endpoints) between two points on an elliptic curve. The sum of those two points is then the negative of the third intersection of that secant on the curve (which is found by flipping a point over horizontal axis). For adding a point to itself, you instead use the tangent at that point. The additive zero for this group is the point at infinity (this is when you have a vertical line that doesn't intersect the graph anywhere else).

pointwise additionpointwise doubling

Purple Alien Planet has a good explanation of point arithmetic (and is also where I got the images). Wolfram Mathworld also has a page on elliptic curves.

Others include the discrete versions (used for elliptic curve cryptography), and although they don't lend themselves to visualization quite as well as the continuous ones, they follow similar principles, and that is what the definition in (b) is referring to.

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  • $\begingroup$ @ AJMansfield: What about a possible bijection! $\endgroup$ – DER Sep 3 '13 at 15:28
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    $\begingroup$ @AR1 Well, the both mean the same thing - the chord is just a line between two points on a curve, and a tangent is the line that is tangent to the curve at our point. The Chord and Tangent law is just a more general version of what I described above regarding how one performs pointwise addition. $\endgroup$ – AJMansfield Sep 3 '13 at 15:32
  • $\begingroup$ @ AJMansfield: I mean bijection between the two cases (a) and (b). $\endgroup$ – DER Sep 3 '13 at 15:34
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    $\begingroup$ @AR1 What are you even looking for here? They are the same! $\endgroup$ – AJMansfield Sep 3 '13 at 15:36
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I am very confused about the group structure you intend to impose on chords and tangents, but here is a basic definition.

A chord

Given a curve $g(x)$, a chord between the points $a$ and $b$ is a line including the points $(a,g(a))$ and $(b,g(b))$. In the case illustrated above, $g(x)=x^2$, $a=0$ and $b=2$. Given a curve $g(x)$, any line intersecting $g(x)$ at least twice is a chord of $g(x)$.

Tangent line

Given a differentiable curve $g(x)$, the tangent line at $c$ is the line intersecting the curve $g(x)$ at $(c,g(c))$ whose slope is $g'(c)$.

Note that these definitions are neither mutually exclusive nor exhaustive. That is, we can have lines that are neither chords nor tangents of $g(x)$

Neither

and lines that are both chords and tangents of $g(x)$

Both

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