In PG(2, q), q a prime power, we study the set T of Baer subplanes that contain a fixed triangle PQR. To construct a linear rank 2–geometry over T , we determine the dihedral groups, their orders and possible extensions that are generated by the involutions of two Baer subplanes of T . If q+1 is an odd prime, the (q+1)2 Baer subplanes through the triangle PQR are the points of an affine plane A...