Investigating centres of cyclic and tangential quadrilaterals coinciding with centroids

In this paper, we investigate cyclic and tangential quadrilaterals with the property that the vertex centroid or the area centroid coincides with the circumcentre and the incentre, respectively, making altogether four cases. However, these four cases are not analogous regarding the variety of quadrilaterals having the claimed property. In one case, there is only the trivial solution. In three cases, there are also other solutions, namely special quadrilaterals with special symmetries. And in one case, there are even many quadrilaterals without any symmetry having the claimed property. We restrict ourselves to elementary investigations, for more advanced considerations we only give corresponding references for interested readers, such that high school students can also follow the argumentations.