Consider four points in the plane, three of which form a triangle. The lengths of the three triangle sides, as well as the lengths of the three segments connecting the triangle vertices to the fourth point, are related as below.[br][br][i]You can drag around any of the points in the diagram.[/i]

This theorem, known to Japanese mathematicians as [i]rokushajutsu[/i] (六斜術), can be proven using the [url=https://www.geogebra.org/m/tn4uebgj]Law of Cosines[/url]. An interesting special case is when [i]x = y = z,[/i] which makes the red point coincide with triangle's circumcenter. Substituting [i]R [/i]for each instance of those variables in the algebra above and solving, one obtains a formula for the circumradius of a triangle in terms of its side lengths:

[center][math]R=\sqrt{\frac{a^2b^2c^2}{2a^2b^2+2a^2c^2+2b^2c^2-a^4-b^4-c^4}}[/math][/center]

The six segment theorem can also be applied for an efficient proof of [url=https://www.geogebra.org/m/ntx5k5yj]Descartes' Circle Theorem[/url]. For more information, see [url=https://www.sangaku-journal.com/2024/SJM_2024_1-8_Unger.pdf]Notes on the Six-Segment Theorem[/url] by J. Marshall Unger.