Orthogonality as well as Equidistance Can Be Used as the Sole Primitive Notion for Euclidean Geometry
Orthogonality as well as Equidistance Can Be Used as the Sole Primitive Notion for Euclidean Geometry
Pieri has shown that the ternary relation of a point equally distant from two other points and (in symbols, ) can be used as the primitive foundation of Euclidean geometry of two or more dimensions [1].
Y
X
Z
XY=YZ
This Demonstration shows that the relation that is the midpoint of can be defined using the relation that , and form a triangle with a right angle at . The definition is
mid
X
YZ
τ
X
Y
Z
X
mid(X,Y,Z)⇔((X=Y∧X=Z)∨∃U∃V(τ(U,Y,Z)∧τ(Y,U,V)∧τ(V,Y,Z)∧τ(Z,V,U)∧τ(X,Y,V)∧τ(X,V,Z)∧τ(X,Z,U)∧τ(X,U,Y)))
Pieri's relation can be defined by
XY=YZ⇔∃R(mid(Y,X,Z)∨(mid(R,X,Z)∧τ(R,X,Y))
Thus, Euclidean geometry can also use the relation as a simple foundation.
τ
You can drag the points and Z, shown as locators.
Y