García Reflection Triangle

The García reflection triangle of a nondegenerate reference triangle is the triangle obtained by reflecting each excenter in the midpoint of the corresponding side. These are point reflections, so in radius vectors,

			(1)
			(2)
			(3)

where , , and are the excenters opposite , , and , respectively (Dalcín and Kiss 2021).

Writing , , and , its vertices have homogeneous barycentric coordinates given by the rows of

(4)

The row sums are , , and . Dividing each row by its sum gives a matrix with determinant 1, so the García reflection triangle has the same area as the reference triangle (Dalcín and Kiss 2021).

The vertex is also the orthocenter of , where is the incenter, with the other two vertices obtained cyclically. Equivalently, the García reflection triangle is the anticevian triangle of with respect to the contact triangle (Dalcín and Kiss 2021). The triangle centroid of is the midpoint of the centroids of the García reflection triangle and the excentral triangle (Dalcín and Kiss 2021).

It is perspective to the outer García triangle at the Nagel point (Dalcín and Kiss 2021). It is also perspective to the Fuhrmann triangle at the Nagel point in the nondegenerate case (Kimberling).

The inner García triangle uses reflections of the incenter in the perpendicular bisectors of the sides.