First, split your L-shape into two neat rectangles. Find the centre of each rectangle—where the diagonals cross, beta. Then, treat each centre as if all the mass of its rectangle is there, and find the balancing point between these two weighted points. Simple.
In my work, finding the balance is everything. A blouse cut even a centimetre off-centre will not give the wearer any 'sukh'. The maths is the same, whether for a lamina or for a perfect-fitting choli.
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