The reason for the validity of this construction is explained in Arbelos.html . The incircle D(DN) is the inverse of the circle with diameter GF with respect to a certain inversion with center at A. Symmetrically it is also the inverse of the circle with diameter IH with respect to a certain inversion with center at B. Hence the center D must lie on the intersection of AK and BJ.

This construction together with the three pictures below are discussed in:

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