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Distance between Two Fermat Points

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Let and be the Fermat points [1] of the triangle ; is the point with minimum sum of distances to the vertices. Let be the circumradius of and be the Brocard angle [2] of .

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The triangles and are the third vertices of equilateral triangles drawn on the outside and inside of .

Define:

(shown in green),

,

.

Then

.

You can drag the vertices , , .

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Contributed by: Minh Trinh Xuan (August 2022)
Open content licensed under CC BY-NC-SA

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References

[1] E. W. Weisstein. "Fermat Points" from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/FermatPoints.html (Wolfram MathWorld).

[2] E. W. Weisstein. "Brocard Angle" from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/BrocardAngle.html (Wolfram MathWorld).

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