Distance between Two Vecten Points

Let [1] and [2] be the inner and outer Vecten points [3] of the triangle . The Vecten points are based on squares constructed on each side of a triangle. Here, 45° isosceles right triangles on each side are shown. Let be the circumradius of and be the Brocard angle [2] of .
Define:
,
,
.
Then
.

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DETAILS

From the above formula we have:
.
References
[1] C. Kimberling. "X(485) = Vecten Point," Encyclopedia of Triangle Centers - ETC. (Jun 6, 2022) faculty.evansville.edu/ck6/encyclopedia/ETC.html#X485.
[2] C. Kimberling. "X(486) = Inner Vecten Point," Encyclopedia of Triangle Centers - ETC. (Jun 6, 2022) faculty.evansville.edu/ck6/encyclopedia/ETC.html#X486.
[3] E. W. Weisstein. "Vecten Points" from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/VectenPoints.html.
[4] E. W. Weisstein. "Brocard Angle" from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/BrocardAngle.html.
[5] M. T. Xuan. "Conway Triangle Notation" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/ConwayTriangleNotation.
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