Dao's six circumcenter theorem is a theorem about the circumcenters of six triangles formed from consecutive sides of a cyclic hexagon. The lines joining opposite pairs of these circumcenters are concurrent, with parallel lines allowed as concurrence at a point at infinity.
More precisely, let 
, ..., 
be six distinct points on a circle, and take subscripts modulo 6. Let 
be the intersection of
the lines 
and 
, and let 
be the circumcenter of
.
Assume that the intersections are finite and that
each pair 
,
consists of distinct points. Then 
, 
, and 
are concurrent or parallel.
Dergiades (2014) gives a proof using complex numbers, and Cohl (2014) and Nguyen (2017) give synthetic proofs.
For a coordinate expression, normalize the given circle to the unit circle in the complex
plane, and let 
and 
be the complex coordinates
of 
and 
,
respectively. The circumcenter formula of Dergiades
(2014, p. 244) gives
 |
Put 
,
,
and 
.
When ![Im[uv^_]!=0](/images/equations/DaosSixCircumcenterTheorem/Inline20.svg)
,
the intersection 
of 
and 
has complex coordinate
![p=q_1+u(Im[wv^_])/(Im[uv^_]),](/images/equations/DaosSixCircumcenterTheorem/NumberedEquation2.svg) |
where 
denotes the imaginary part and the bar denotes
the complex conjugate. If this denominator vanishes,
another nonparallel pair of the three joining lines may
be used. For an original circle with complex center 
and radius 
, use 
, where 
is the original complex coordinate
of 
.
The original complex coordinate of 
is then 
.
