# Equilateral Triangle Bisection

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 $$\int_{-\pi/6}^{\pi/6}\frac12 r^2 d\theta = \Delta/2$$

• Do you only need an example or you need more? – Luca Goldoni Ph.D. Jan 3 at 9:31
• Derivation of DE of at least one partitioning smooth curve. – Narasimham Jan 3 at 14:26
• OK one possibility is we can start with y = - kx^2 through center and adjust k. – Narasimham Jan 3 at 14:32
• 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. – Luca Goldoni Ph.D. Jan 3 at 15:59
• Please post it as DE can be always found from them. – Narasimham Jan 3 at 19:27

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 $$F\left( x \right) = \left\{ \begin{gathered} 0\,\,\,\,\,if\,\,\,\,x \in \left[ {x_A ,x_E } \right] \hfill \\ f_L (x)\,\,\,if\,x \in \left( {x_E ,x_F } \right) \hfill \\ 0\,\,\,if\,\,\,\,x \in \left[ {x_F ,x_L } \right] \hfill \\ f_R (x)\,\,\,if\,x \in \left( {x_L ,x_M } \right) \hfill \\ 0\,\,\,\,\,if\,\,\,\,x \in \left[ {x_M ,x_H } \right] \hfill \\ \end{gathered} \right.$$
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 $$G$$.