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In a circle, if we pick any two distinct points $p_1$ and $p_2$ and draw a line passing through $p_1$ and $p_2$ that line is parallel to the line tangent to the circle at the midpoint of the arc with endpoints $p_1$ and $p_2$.

Are there curves other than cicles such that is true? What about parabolas?

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  • $\begingroup$ parallel to "the line tangent to the circle at the midpoint of $p_1$ and $p_2$", what do you mean? which circle? $\endgroup$
    – MathFail
    Commented Aug 30, 2022 at 3:55
  • $\begingroup$ "In a circle..." That circle, @MathFail What circle could OP mean? $\endgroup$ Commented Aug 30, 2022 at 3:57
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    $\begingroup$ Well, a line satisfies this condition, of course. But a line can be thought of as a degenerate circle. $\endgroup$ Commented Aug 30, 2022 at 3:59
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    $\begingroup$ How are you defining the midpoint of the curve between the two points? $\endgroup$
    – ConMan
    Commented Aug 30, 2022 at 4:34
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    $\begingroup$ The parabola $f(x)=x^2$ satisfies that for $p_1=(a,a^2)$ and $p_2=(b,b^2)$ the tangent parallel to the secant through $p_1$ and $p_2$ is the tangent at x$x = (a+b)/2$. $\endgroup$ Commented Aug 30, 2022 at 18:08

2 Answers 2

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In my opinion, this is a great question with a clear interpretation. I don't have an answer yet, but I can at least explain how to formulate the question precisely, which may help others find a solution.

Let's suppose the curve is a regular smooth plane curve parametrized by $\mathbb{R}$. Then without loss of generality it may be parametrized by arc-length, i.e. we have a smooth function $\gamma : \mathbb{R} \to \mathbb{R}^2$ such that $\lVert \gamma'(t) \rVert = 1$ for all $t \in \mathbb{R}$.

The condition that the secant line be parallel to the tangent at the midpoint then says that there exists a function $c : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ such that $$\gamma(b) - \gamma(a) = c(a,b) \gamma'\left(\frac{a+b}{2}\right)$$ for all $a,b \in \mathbb{R}$. Using the fact that $\gamma'$ is a unit vector, we have $$c(a,b) = c(a,b) \left\lVert \gamma'\left(\frac{a+b}{2}\right) \right\rVert^2 = c(a,b) \gamma'\left(\frac{a+b}{2}\right) \cdot \gamma'\left(\frac{a+b}{2}\right) = (\gamma(b) - \gamma(a)) \cdot \gamma'\left(\frac{a+b}{2}\right).$$ We wish to know if this forces $\gamma$ to be a circle or a line, i.e. a curve of constant curvature. So the question can be posed as:

Let $\gamma : \mathbb{R} \to \mathbb{R}^2$ be a smooth function such that $$\lVert \gamma'(a) \rVert = 1$$ and $$\gamma(b) - \gamma(a) = \left((\gamma(b) - \gamma(a)) \cdot \gamma'\left(\frac{a+b}{2}\right)\right) \gamma'\left(\frac{a+b}{2}\right)$$ for all $a, b \in \mathbb{R}$. Must $t \mapsto \lVert \gamma''(t) \rVert : \mathbb{R} \to \mathbb{R}$ be constant?

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Improved answer

Let $\gamma$ be immersive. As $\gamma(s+h)-\gamma(s-h)$ is parallel to $\gamma'(s)$, we have $$\gamma(s+h)-\gamma(s-h)=\langle\gamma(s+h)-\gamma(s-h),\gamma'(s)/\|\gamma'(s)\|\rangle \frac{\gamma'(s)}{\|\gamma'(s)\|}.$$ Now take the second derivative in respect to $h$ to get $$\gamma''(s+h)-\gamma''(s-h)=\langle\gamma''(s+h)-\gamma''(s-h),\gamma'(s)/\|\gamma'(s)\|\rangle \frac{\gamma'(s)}{\|\gamma'(s)\|}.$$ Divide by $2h$ and let $h\to0$: $$\gamma'''(s)=\langle\gamma'''(s),\gamma'(s)/\|\gamma'(s)\|\rangle\gamma'(s)\frac{\gamma'(s)}{\|\gamma'(s)\|},$$ that is, $\gamma'''(s)$ is parallel to $\gamma'(s)$,

In this case $$\left( \det(\gamma',\gamma'') \right)'=0,$$ hence $$\kappa(s)=\frac{\det\bigl(\gamma'(s),\gamma''(s)\bigr)}{\|\gamma'(s)\|^3} =\frac{\text{constant}}{\|\gamma'(s)\|^3},$$ that is, the curvature solely depends on the length of the velocity vector. For an arc-length parametrisation it follows that the curvature is constant.

As example would serve $\gamma(s)=(a\cdot e^s,b\cdot e^{-s})$ for constants $a$ and $b$.

Original answer

As $\gamma'(s)$ is a unit vector and $\gamma(s+h)-\gamma(s-h)$ is parallel to $\gamma'(s)$, we have $$\gamma(s+h)-\gamma(s-h)=\langle\gamma(s+h)-\gamma(s-h),\gamma'(s)\rangle\gamma'(s),$$ as @diracdeltafunk remarked.

Now take the second derivative in respect to $h$ to get $$ \gamma''(s+h)-\gamma''(s-h)= \langle\gamma''(s+h)-\gamma''(s-h),\gamma'(s)\rangle\gamma'(s). $$ Divide by $2h$ and let $h\to0$: $$\gamma'''(s)=\langle\gamma'''(s),\gamma'(s)\rangle\gamma'(s),$$ that is, $\gamma'''(s)$ is parallel to $\gamma'(s)$, hence the derivative of $\|\gamma''(s)\|$ vanishes: $$ \begin{align} \frac{d}{ds}\|\gamma''(s)\|^2&=2\langle\gamma''(s),\gamma'''(s)\rangle\\ &=2\langle\gamma''(s),\langle\gamma'''(s),\gamma'(s)\rangle\gamma'(s)\rangle\\ &=2\langle\gamma'''(s),\gamma'(s)\rangle\langle\gamma''(s),\gamma'(s)\rangle\\ &=\langle\gamma'''(s),\gamma'(s)\rangle \frac{d}{ds} \lVert \gamma'(s) \rVert^2\\ &=0. \end{align} $$

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