Dao's six point circle

Dao's six point circle (blue) through six centers $A_{B}$, $A_{C}$, $B_{C}$, $B_{A}$, $C_{A}$, $C_{B}$

In Euclidean geometry, Dao's six point circle is a circle that contains the centers of six special circles associated with an arbitrary triangle. Each of those circles is tangent to one of the medians of the triangle at its centroid, and goes through one of the two vertices that are not on that median.

Namely, if $A$, $B$, and $C$ are the vertices of the triangle, $AM_{a}$, $BM_{b}$ and $CM_{c}$ are its medians, and $G$ is its centroid, then the six circles are $A_{B}$ and $A_{C}$, that go through $A$ and are tangent to $BM_{b}$ and to $CM_{c}$, respectively; $B_{C}$ and $B_{A}$, that go through $B$ and are tangent to $CM_{c}$ and $AM_{a}$, respectively; and $C_{A}$ and $C_{B}$, that go through $C$ and are tangent to $AM_{a}$ and $BM_{b}$, respectively.

The fact that the centers of those six circles lie on a common circle was noted in November 2013 by Đào Thanh Oai. García Capitán provided a formula for the radius of the common circle, and generalized the result to three concurrent cevians, in which case the six centers lie on a conic section. The center of Dao's circle is triangle center number $X(5569)$ in Clark Kimberling's list. The circle is related to the Van Lamoen circle as generalized by A. Myakishev.

References

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📰 Article(s) of the same category(ies)

• Derivation of the Cartesian form for an ellipse
• Dao's theorem
• Abouabdillah's theorem

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