Compass and straightedge contruction of an equilateral triangle inscribed in a given circle with unknown center I haven't found a proper solution for this problem, found in Hartshorne's "Geometry: Euclid and Beyond": 

(4.3) Given a circle, but not given its center, construct an inscribed equilateral triangle in as few steps as possible. 

I managed to construct it in $9$ steps (use of compass or straightedge) but I can't get any lower. Finding circle center takes $5$ of those $9$ uses, and then I need $1$ more to get vertices and $3$ for constructing the triangle.

This problem can be found in the online/android/ios puzzle game Euclidea. I came across this thread while researching alternative solutions.
As constructing points doesn't count as a step, the most efficient solution I have found is six steps.
Given circle $c1$ with unknown center:


*

*Mark arbitrary points $P1$ and $P2$ on $c1$

*Draw circle $c2$ with center $P1$ and radius $\overline{P1P2}$

*Mark point $P3$ at the intersection of $c1$ and $c2$

*Draw circle $c3$ with center $P2$ and radius $\overline{P2P3}$

*Mark point $P4$ at the intersection of $c2$ and $c3$

*Draw circle $c4$ with center $P4$ and radius $\overline{P4P2}$

*Mark points $P5$ and $P6$ at the intersections of $c3$ and $c4$

*Draw line through $P5$ and $P2$, intersecting $c1$ at $P7$

*Draw line through $P6$ and $P2$, intersecting $c1$ at $P8$

*Draw line segment $\overline{P7P8}$ to complete $\triangle P2P7P8$



