Van Lamoen circle

The circle through the centroids of the six triangles formed when three cevians through one point divide a triangle. Its center is the centroid of the original triangle.

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Cevian: A line segment from a vertex of a triangle to the opposite side or its extension. The three cevians through one point create the six triangles whose centroids define the circle.
Triangle: A polygon with three sides and three vertices. The construction begins with a triangle and depends on its vertices and opposite sides.
Floor van Lamoen: A Dutch mathematician known for work and problem posing in triangle geometry. The circle bears his name through a problem about centroids of cevian subtriangles.
Van Lamoen circle theorem: The result that the six centroids of cevian subtriangles lie on a circle centered at the original triangle’s centroid. This theorem states the defining geometric property and its center.
Centroid: The point where a triangle’s three medians meet, balancing its area uniformly. Each of the six defining points is a subtriangle centroid, and their circle is centered at the original centroid.
Triangle center: A point defined from a triangle by a geometric or algebraic construction. The original triangle’s centroid is the center of the Van Lamoen circle.
Forum Geometricorum: An international journal devoted to geometry, especially classical and elementary geometry. The circle’s properties were publicized in the modern triangle-geometry literature associated with this journal.
Symmedian: A cevian isogonal to a triangle’s median, meeting the opposite side at a point determined by squared side lengths. Choosing concurrent symmedians gives a structured special case of the cevian-centroid circle.
Ceva's theorem: A theorem giving a condition for three cevians of a triangle to be concurrent. The shared cevian point that partitions the triangle is governed by cevians’ concurrency.
Ceva's configuration: A triangle together with three concurrent cevians, one from each vertex. This is the basic arrangement that divides the triangle into six subtriangles.