= Euclidean geometry
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<Euclidean geometry> describes flat <Euclidean space> using <lengths>, <angles>, and <rigid motions>. A coordinate realization uses points $x,y\in\mathbb R^n$, <Euclidean distance>
$$
d(x,y)=\sqrt{\sum_{i=1}^n(x_i-y_i)^2},
$$
and <angles> determined by the <dot product>. The coordinate axes are a convenience: <translations> and <rotations> change coordinates while preserving these measurements. This makes <Euclidean geometry> useful for comparing figures and converting measurements along different directions into a single distance.
In the <Euclidean plane>, through a point outside a line there is exactly one parallel line. This is a standard equivalent formulation of the <parallel postulate>. Euclid's fifth postulate in the <Elements> instead describes when two lines meet using the interior <angles> made by a crossing line.https://mathcs.clarku.edu/~djoyce/elements/bookI/post5.html[Euclid, Elements I, Postulate 5]{ref} The assumption has consequences we often take for granted: a <triangle>'s interior <angles> sum to $\pi$, and <similar triangles> can differ in size while keeping their <angles>. Comparing with <spherical geometry> reveals that these are properties of a particular geometry.
As a computed example, the diagonal from $(0,0)$ to $(4,3)$ in a <rectangle> has <length> $\sqrt{4^2+3^2}=5$. The <Pythagorean theorem> explains why perpendicular contributions add as squares. If the directions cease to meet at a <right angle>, the <law of cosines> supplies the missing <angle>-dependent term. The beauty is that a single distance formula unites geometric constructions, coordinate calculations, and <orthogonality> in <linear algebra>.
Make the assumptions visible: in a <Cartesian coordinate system> with <orthogonal> axes and the same <length> unit on each axis, take $A=(0,0)$, $B=(4,0)$ and $C=(0,3)$. The <vectors> $B-A=(4,0)$ and $C-A=(0,3)$ have <dot product> zero, establishing a <right angle> at $A$ before using the <Pythagorean theorem>. A sketch that merely looks like a <right triangle> supplies no such guarantee. Likewise, an image stretched horizontally and vertically by different amounts can distort both <lengths> and <angles>; a displayed shape and the assumptions defining it must be kept separate.
For a less origin-dependent measurement, take $P=(1,2)$ and $Q=(5,5)$. Their displacement <vector> is $Q-P=(4,3)$, so their <Euclidean distance> is again $5$. Its <Euclidean norm> records the direct separation, whereas travelling first horizontally and then vertically covers <length> $4+3=7$. This connects measuring a diagonal to locating two positions from a <Cartesian coordinate system>. A <translation> changes where the positions are recorded without changing their separation; a <rotation> changes the component values without changing the <Euclidean norm>.
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