The area enclosed by the envelopes of two families of lines in a triangle

We compute the relative area of the envelope of all lines dividing the area of any triangle in ratio p/(1−p), and the relative area of its right perimeter analogue, the envelope of lines that divide the perimeter of an equilateral triangle in the same ratio, both for 0<p≤1/2. These expressions generalise their known particular values of −1/2+(3/4)ln⁡(2) and 1/12 for p=1/2. For area p-envelopes, we provide a shorter second proof to a result by Russell Gordon on the incidence partitions of these envelopes. The perimeter p-envelopes are a new family of curves and so is the entire work related to them. Both area computations are based on ideas from calculus, inclusion-exclusion, affine transformations and dihedral groups.