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My first instinct was to approach this in terms of analytic curves (say, in the projective plane) and their dual curves. But this is total overkill. It should just be an elementary exercise in the differential geometry of curves, if we make some reasonable assumptions. When $X$ is contained in a line, its curvature $\kappa$ is identically $0$. Real ...

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As commenters pointed out, any smooth curve with 4-fold symmetry is a counterexample, such as $\gamma = \{(x,y): x^4+y^4 = 1\}$. Indeed, 4-fold symmetry means there is a point $C$ such that rotation by $90$ degrees about $C$ maps $\Gamma$ to itself. Consequently, such rotation maps the center of mass $O$ to itself, which implies $O=C$. Since rotation ...

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The vector$\overrightarrow{AB}=(3,3,3)$, hence the vector $(1,1,1)$, is normal to the perpendicular bisector plane. Thus its equation is $\;x+y+z=d$. Furthermore, it passes through the midpoint of $(AB)$ which has coordinates $\;\Bigl(\dfrac52,\dfrac72,\dfrac92\Bigr)$. Finally the equation is: $$x+y+z=\frac{21}2.$$

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Differentiating two times the equation of the ellipse leads to the quations which are solved as shown in attachment.

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In dynamics, time taken for rolling oscillation of a small heavy marble irrespective of amplitude in such a shaped trough.. is constant $( = 2 \pi \sqrt {\frac{4 a}{g}})$.. Tautochrone property. EDIT1: Distance of any cycloid point to x-axis ( on which the circle rolls) along its normal is half the radius of its curvature...one of its properties.

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The Brachistochrone curve between two points at the same height is a cycloid.

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