# What are Euclidean (Euclidean) Relations?

June 26, 2021Euclid’s five axioms are:

– One and only one line passes through two points.

– A line segment can be extended indefinitely in both directions.

– A circle can be drawn with a center and a point on it.

– All right angles are equal.

– One and only one parallel can be drawn from a point taken outside of a line.

Height Relations

In a right triangle, the square of the height of the hypotenuse is equal to the product of the 2 sides it separates on the hypotenuse.

Its equation is “h.h=k.p”.

Perpendicular Relation

In a right triangle, the square of the length of a right side is equal to the product of the perpendicular projection of that side on the hypotenuse and the length of the hypotenuse. This relation is called Euclid’s Right Side Relation.

a) B/k = a/b =» k.a is found from ABC ~ HAC similarity.

b) ABC ~ HBA analogy gives c/p =a/c =» p.a.

Using Euclidean relations, 1/h² = 1/b² + 1/c², using ABC ~ HBA similarity. It can be shown that b.c = h.a.