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Orthogonality as well as Equidistance Can Be Used as the Sole Primitive Notion for Euclidean Geometry

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Pieri has shown that the ternary relation of a point
Y
equally distant from two other points
X
and
Z
(in symbols,
XY=YZ
) can be used as the primitive foundation of Euclidean geometry of two or more dimensions [1].
This Demonstration shows that the relation
mid
that
X
is the midpoint of
YZ
can be defined using the relation
τ
that
X
,
Y
and
Z
form a triangle with a right angle at
X
. The definition is
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
τ
as a simple foundation.
You can drag the points
Y
and Z, shown as locators.
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