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  "body": "\n## Definition\n\nThe language has variables for points and two relations.\nThe ternary relation $B(x, y, z)$ reads \"$y$ lies [[between|CE8GX6ADZS]] $x$ and $z$\".\nThe quaternary relation $xy \\equiv zw$ reads \"the segment $xy$ is congruent to the segment $zw$\".\n*Tarski's axioms* are the following first-order sentences, each universally closed.\n\n1. Reflexivity of congruence: $xy \\equiv yx$.\n2. Identity of congruence: if $xy \\equiv zz$, then $x = y$.\n3. Transitivity of congruence: if $xy \\equiv zu$ and $xy \\equiv vw$, then $zu \\equiv vw$.\n4. Identity of betweenness: if $B(x, y, x)$, then $x = y$.\n5. Pasch: if $B(x, u, z)$ and $B(y, v, z)$, then there is $a$ with $B(u, a, y)$ and $B(v, a, x)$.\n6. Continuity, a schema: let $\\varphi(x)$ and $\\psi(y)$ be formulas in which $a$ and $b$ are not free. If some $a$ has $B(a, x, y)$ whenever $\\varphi(x)$ and $\\psi(y)$, then some $b$ has $B(x, b, y)$ whenever $\\varphi(x)$ and $\\psi(y)$.\n7. Lower dimension: there are $a, b, c$ such that none of $B(a, b, c)$, $B(b, c, a)$, $B(c, a, b)$ holds.\n8. Upper dimension: if $xu \\equiv xv$, $yu \\equiv yv$, $zu \\equiv zv$ and $u \\ne v$, then $B(x, y, z)$ or $B(y, z, x)$ or $B(z, x, y)$.\n9. Euclid: if $B(x, u, t)$, $B(y, u, z)$ and $x \\ne u$, then there are $v, w$ with $B(x, y, v)$, $B(x, z, w)$ and $B(v, t, w)$.\n10. Five segments: if $x \\ne y$, $B(x, y, z)$, $B(x', y', z')$, $xy \\equiv x'y'$, $yz \\equiv y'z'$, $xu \\equiv x'u'$ and $yu \\equiv y'u'$, then $zu \\equiv z'u'$.\n11. Segment construction: for all $x, y, a, b$ there is $z$ with $B(x, y, z)$ and $yz \\equiv ab$.\n\n## Remarks\n\nExample: the plane $\\R^2$, with $B(x, y, z)$ meaning that $y$ is on the closed segment $[x, z]$ and $xy \\equiv zw$ meaning $\\abs{x - y} = \\abs{z - w}$, satisfies all eleven axioms.\nNon-example: the line $\\R$ with the same relations fails the lower dimension axiom, since of any three reals one lies between the other two.\nThe models are exactly the planes $K^2$ over [[real closed fields|DDY3Q3PJ6W]] $K$, with the relations defined as in $\\R^2$.\nThe [[theory|92SWXST1ZM]] they define is [[complete|655K8Y7VS8]] and decidable.\nAxiom 9 is a form of the parallel postulate; without it one gets an axiom system for absolute geometry.\n",
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