right trapeziums

the Ancient Babylonians (up to 4000 years ago) appear to have worked substantially with right-angled triangles and right angled trapeziums

one of their many clever techniques is that of being able to split a right-angled trapezium into two equal areas (seemingly for inheritance purposes - with fields and orchards)

in a study reported early in 2016, Mathieu Ossendrijver claimed to show that the Babylonians had tracked the path of Jupiter

one of the techniques involved in this calculation involved splitting a right-trapezium into two equal areas

splitting a right trapezium into two equal areas is a difficult problem

initially, consider the simpler problem of finding the length of a line part way along a right-trapezium

using 'steps': the Babylonians had an understanding of (what gets translated as) 'feed': how much you go down (or up) for a certain distance across
(i.e. the tan of the angle of depression)

in some of these questions the line does split the area in two

which?

for a base split in the ratio 1 : 2
how do you find the length 'n' ?

how do you do this in general?

turning to the much harder problem of how to dissect the area, just for the case where the base ratio is 1 : 2

using the result previously obtained for the distance 'n' being the weighted mean of 'a' and 'b' in the ratio 1 : 2

using the areas of the trapeziums
left = right

(or you could also use e.g. left = half of the whole)