The lines which bisect the area of a triangle form an envelope as shown in this picture

It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is $$\frac{3}{4} \log_e(2) - \frac{1}{2} \approx 0.01986.$$

But this is also $$\sum_{n=1}^{\infty}\frac{1}{(4n-1)(4n)(4n+1)}.$$

Is there any connection between the series and the deltoid? Or is it just a coincidence?