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.

Two circles + five lines = 7 steps:



*

*Select an arbitrary point $P_0$ on a given circle $C_0$ 
and draw a circle $C_1$
with an arbitrary radius $r$, which is small enough to intersect $C_0$.

*Draw a circle $C_2$ centered at the intersection point $P_1$ with the same radius $r$
to get the intersection points $P_2$ and $P_3$.

*Draw a line through points $P_2,P_1$, intersecting $C_0$ at point $P_4$.

*Draw a line through points $P_1,P_3$, intersecting $C_0$ at point $P_5$.

*Draw a side $P_5 P_4$ of the inscribed equilateral triangle.

*Draw a side $P_4 P_0$ of the inscribed equilateral triangle.

*Draw a side $P_0 P_5$ of the inscribed equilateral triangle.
