Centroids of a Quadrilateral

We discussed centroids a lot in the first half of Linear Algebra, and also we discussed Varignon parallelograms a lot during that time and in that course.

Besides being desparate Linear Algebra dynamic worksheet topic, I got curious about what happens if we do the four centroids of the four interior triangles of a quadrilateral. Therefore, I put this together.

I set up the quadrilateral as an irregular quadrilateral. Points A, B, C, D can be manipulated. I have the centroids of Triangles ABC, ADC, ABD, BCD and a quadrilateral made out of those. The angle measures are labeled. I also provided the centroid of Quadrilateral ABCD as point G.

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The quadrilateral made out of the centroids is similar to Quadrilateral ABCD. Compare the angles at opposite sides.

Transformation Geometry Note: This looks like a signed dilation over G with scaling factor -1/2.

Darron Steele, Created with GeoGebra