Extension: Geometric Interpretation of Cauchy's Mean Value Theorem

Previously, we talked about Rolle's Theorem, which states that given a function $f(x)$ continuous on $[a,b)$ and differentiable on $(a,b)$, if $f(a)=f(b)$ then there exists a constant $c$ in $(a,b)$ such that $$f'(c)=0.$$ Cauchy's Mean Value Theorem is a natural generalization of Rolle's Theorem (and also the Mean Value Theorem), involving two functions. It specifically states the following:

Cauchy's Mean Value Theorem: Suppose functions $f(x), g(x)$ are continuous on $[a,b]$ and differentiable on $(a,b)$. Then there exists $c$ in $(a,b)$ such that $$[f(b)-f(a)]g'(c)=[g(b)-g(a)]f'(c).$$ Or, if $g(a)\ne g(b)$ and if $g'(c)\ne 0$, then we can also say $$\frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}.$$ Okay, so this resembles the regular Rolle's Theorem and Mean Value Theorem algebraically, but what does this mean geometrically? Well, we get a hint from the previous section on "parametric" Rolle's Theorem, which suggests that we consider the two functions $f(x)$ and $g(x)$ as parameters: i.e. setting $x=f(t)$ and $y=g(t)$. Using this idea, see if you can visualize this theorem geometrically:

Any ideas? Let's work on this together: suppose we plot the graph of the curve parametrized by $x=f(t)$ and $y(t)$. One example is shown below:

We notice that the right hand side of the equation above is imply the expression of the slope of the line containing $(f(a),g(a))$ and $(f(b),g(b))$. So geometrically, the theorem tells us that there is a value $c$ in $(a,b)$ for which the tangent line to the curve at $(f(c),g(c))$ is parallel to the line connecting the two endpoints.