Equivalently, given length a=BD and b=AB, find c and d so that a/c = c/d = d/b The construction involves Nicomedes Conchoid, and as a last step you need to slide point K so that it lies on the red Conchoid curve.
Look at the ratios BD:DK, DK:AL, AL:AB. You should notice that two stay the same regardless of where K lies. Why? When K lies on the Conchoid the ratios should all be approximately the same. How does the ratio relate to the length BD and AB? You can try different values for BD and AB, but BD needs to be bigger than AB or the construction collapses.