Triangle 1 (see the picture) is given. Find the point toward which the vertices of triangle n -> infinity converge, assuming that triangle n is constructed by uniting the feet of the altitudes of triangle n-1.
Sequence of triangles formed by the above mentioned rule.
For the definition of "foot of an altitude" please see: Perpendicular Foot
By denoting as $T_n$ the triangle at the $n$-th iteration we can easily describe the angles of $T_{n+1}$, orthic triangle of $T_n$, in terms of the angles of $T_n$. We may check that the area and the perimeter of $T_n$ converge to zero, but the "shape" of $T_n$ (i.e. the triple of the angles) does not converge, in the general case.
Actually it is known that such iteration is usually chaotic, and not difficult to prove: assuming that our sequence is convergent to a point $P$, from some $n$ onward the orthocenter of $T_n$ has to lie in the interior of $T_n$, meaning that $T_n$ is acute-angled for any $n$ sufficiently large. On the other hand the shape of $T_n$ changes according to $$(A,B,C)\to (\pi-2A,\pi-2B,\pi-2C) $$ and almost surely the map sending $x$ into $-2x\pmod{\pi}$ is not convergent.