  In this paper, we consider the intrinsic Diophantine approximation on the
triadic Cantor set $\mathcal{K}$, i.e. approximating the points in
$\mathcal{K}$ by rational numbers inside $\mathcal{K}$, a question posed by K.
Mahler. By using another height function of a rational number in $\mathcal{K}$,
i.e. the denominator obtained from its periodic 3-adic expansion, a complete
metric theory for this variant intrinsic Diophantine approximation is presented
which yields the divergence theory of Mahler's original question.
