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Any triangle in an isotropic plane has a circumcircle $u$ and an incircle $i$. It turns out that there are infinitely many triangles with the same circumcircle $u$ and incircle $i$. This one-parameter family of triangles is called a poristic system of triangles.
We prove that all triangles in a poristic system share the centroid and the Feuerbach point. The symmedian point and the Gergonne point of the triangle $P_1P_2P_3$ move on the lines while the triangle traverses the poristic family. The Steiner point of $P_1P_2P_3$ traces a circle, and the Brocard points of $P_1P_2P_3$ trace a quartic curve.
We also study the traces of some further points associated with the triangles of the poristic family. The vertices of the tangential triangle move on a circle while the initial triangle traverses the poristic family, and the centroid of the tangential triangle is fixed.
