## Geometry Loci Doodling

Here's three geometry loci related to cyclic quadrilaterals which I found a few days ago while doodling with the dynamic geometry software, Sketchpad. Use the LINK buttons to go from one sketch to the other.

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } Geometry Loci Doodling

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1) In the 1st sketch above, drag point D. What do you notice about the locus of the point mass centroid of a cyclic quadrilateral?
2) In the 2nd sketch above, drag point D. What do you notice about the locus of the lamina centroid of a cyclic quadrilateral?
3) In the 3rd sketch above, drag point D. What do you notice about the locus of the perimeter centroid 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 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 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.

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Created by Michael de Villiers, 16 August 2020 with WebSketchpad.