Desargues Configuration - Construction From Perspective Triangles

Construction From Perspective Triangles

Two triangles ABC and abc are said to be in perspective centrally if the lines Aa, Bb, and Cc meet in a common point (the so-called center of perspectivity). They are in perspective axially if the crossing points of the lines through pairs of corresponding triangle sides X = AB·ab, Y = AC·ac, and Z = BC·bc all lie on a common line, the axis of perspectivity. Desargues' theorem in geometry states that these two conditions are equivalent: if two triangles are in perspective centrally then they must also be in perspective axially, and vice versa. When this happens, the ten points and ten lines of the two perspectivities (the six triangle vertices, three crossings points, and center of perspectivity, and the six triangle sides, three lines through corresponding pairs of vertices, and axis of perspectivity) together form an instance of the Desargues configuration.