  The local structure of a fractal set is described by its dimension $D$, which
is the exponent of a power-law relating the mass ${\cal N}$ in a ball to its
radius $\epsilon$: ${\cal N}\sim \epsilon^D$. It is desirable to characterise
the {\em shapes} of constellations of points sampling a fractal measure, as
well as their masses. The simplest example is the distribution of shapes of
triangles formed by triplets of points, which we investigate for fractals
generated by chaotic dynamical systems. The most significant parameter
describing the triangle shape is the ratio $z$ of its area to the radius of
gyration squared. We show that the probability density of $z$ has a phase
transition: $P(z)$ is independent of $\epsilon$ and approximately uniform below
a critical flow compressibility $\beta_{\rm c}$, but for $\beta>\beta_{\rm c}$
it is described by two power laws: $P(z)\sim z^{\alpha_1}$ when $1\gg z\gg
z_{\rm c}(\epsilon)$, and $P(z)\sim z^{\alpha_2}$ when $z\ll z_{\rm
c}(\epsilon)$.
