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Please help find different equation of (smooth continuous) curves through the centroid of an equilateral triangle bisecting the triangle area $\Delta$ into two equal parts.

I could not proceed after starting at a vertex $$ \int_{-\pi/6}^{\pi/6}\frac12 r^2 d\theta = \Delta/2 $$

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  • $\begingroup$ Do you only need an example or you need more? $\endgroup$ – Luca Goldoni Ph.D. Jan 3 at 9:31
  • $\begingroup$ Derivation of DE of at least one partitioning smooth curve. $\endgroup$ – Narasimham Jan 3 at 14:26
  • $\begingroup$ OK one possibility is we can start with y = - kx^2 through center and adjust k. $\endgroup$ – Narasimham Jan 3 at 14:32
  • $\begingroup$ Your choice is good even though it is not what you are searching for. I mean it is not the differential equation. I found many other special ways but not the differential equation. If you want I can post a picture with my general idea. $\endgroup$ – Luca Goldoni Ph.D. Jan 3 at 15:59
  • $\begingroup$ Please post it as DE can be always found from them. $\endgroup$ – Narasimham Jan 3 at 19:27
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This is my general idea:

  1. We use two bump functions of the kind $$ f(x) = \left\{ \begin{gathered} be^{ - \frac{1} {{a^2 - x^2 }}} \,\,\,\,if\,\,\,\, - a \leqslant x \leqslant a \hfill \\ 0\,\,\,\,elsewhere \hfill \\ \end{gathered} \right. $$ and $g(x)=-f(x)$.
  2. We translate them on the left and on the right of the centroid and we call them $f_L(x)$ and $f_R(x)$.
  3. We define a new function $F(x)$ as $$ F\left( x \right) = \left\{ \begin{gathered} 0\,\,\,\,\,if\,\,\,\,x \in \left[ {x_A ,x_E } \right] \hfill \\ f_L (x)\,\,\,if\,x \in \left( {x_E ,x_F } \right) \hfill \\ 0\,\,\,if\,\,\,\,x \in \left[ {x_F ,x_L } \right] \hfill \\ f_R (x)\,\,\,if\,x \in \left( {x_L ,x_M } \right) \hfill \\ 0\,\,\,\,\,if\,\,\,\,x \in \left[ {x_M ,x_H } \right] \hfill \\ \end{gathered} \right. $$
  4. The parametric curve obtained from this function satisfy your request.

enter image description here Of course you can add as many couples of bump functions as you want and you can play with parameters freely with parameters $a$ and $b$ as long as the graph of $f$ stay inside the triangle and there is an interval where the function $F$ is zero containing $G$.

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