Here's three geometry loci related to cyclic quadrilaterals which I found a few days ago while doodling with the dynamic geometry software, Sketchpad. Click on the Link to buttons at the bottom to go from one sketch to the other.
Geometry Loci Doodling of the Centroids of a Cyclic Quadrilateral
Explore
(Click on the Link to buttons at the bottom to go from one sketch to the other.)
1) In the 1st sketch above, drag point D. What do you notice about the locus of the point mass centroid Gpm of a cyclic quadrilateral?
2) In the 2nd sketch above, drag point D. What do you notice about the locus of the lamina centroid GL of a cyclic quadrilateral?
3) In the 3rd sketch above, drag point D. What do you notice about the locus of the perimeter centroid Gp of a cyclic quadrilateral?
Challenge
Can you explain why (prove) your observations about the above loci are true?
1) A neat geometric proof of the 1st result appears in a 2016 IJMEST paper by Flavia Mammana from Italy, which can be accessed at: Homothetic transformations and geometric loci: properties of triangles and quadrilaterals.
2) However, by looking at the 1st result from a physics perspective, as pointed out by my Polish colleague, Waldemar Pompe, and considering the point mass centroid as the arithmetic average of the vertices, it is then easy to prove. If |a|=|b|=|c|=|d| represent the vertices of the quadrilateral, then Gpm, denoted by s for simplicity, is s = (a+b+c+d)/4. Thus |s-(a+b+c)/4| = |d|/4, which of course proves the result, and an even more general one:
If D varies on any fixed circle with centre O and radius r (not necessarily on the circumcircle of ABC) then s lies on a circle with radius r/4. Moreover, the centre of the smaller circle is the point mass (vertex) centre of the points A, B, C, and O.
More-over, it generalizes even further to n points (not necessarily on the circle O on which D lies), in which case one would get a smaller circle with radius r/(n+1) with its centre the point mass centre of O and the other n points.
3) An Israeli colleague, Michael N. Fried, has neatly generalized the 1st locus of the point mass centroid of the cyclic quadrilateral ABCD as D is moved on the circle, showing that the obtained locus is merely the result of a homothetic (similar) transformation. Go to: Locus Problem Involving the Intersection of the Side Bisectors of a Quadrilateral.
4) Using elementary geometry, Michael N. Fried has also shown in regard to the 2nd locus that one special case is a strophoid and another special limiting case is a Cissoid of Diocles.
The other two 'doodle' results may or may not be novel, and might require some cumbersome algebra. I'd be happy to hear of any proofs by readers, preferably geometric, if possible.
Related Links
Cyclic Quadrilateral (Rethinking Proof activity)
The Center of Gravity of a Triangle (Rethinking Proof activity)
Point Mass (Vertex) Centroid of Quadrilateral
Centroid of Cardboard (Lamina) Quadrilateral
Centroid of Perimeter Quadrilateral
Three different centroids (balancing points) of a quadrilateral
Investigating Centres of Cyclic & Tangential Quadrilaterals coinciding with their Centroids
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Created by Michael de Villiers, 16 August 2020 with WebSketchpad; updated 25 Nov 2025; 7 March 2026.