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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $S$ be a set of involutions of $G$, that is, of elements of order $2$.\nAssume that $S$ generates $G$ and is closed under conjugation by elements of $G$.\nCall the elements of $S$ *lines*, written $a, b, c, g, h, j$.\nFor $\\sigma, \\tau \\in G$, write $\\sigma \\mid \\tau$ if $\\sigma \\tau$ is an involution.\nCall the elements $ab$ with $a, b \\in S$ and $a \\mid b$ *points*, written $P, Q$.\nFor a point $P$ and a line $g$, $P \\mid g$ means that $P$ lies on $g$.\nFor lines $a$ and $b$, $a \\mid b$ means that they are perpendicular.\nThe pair $(G, S)$ is a *metric plane* if:\n\n1. for any points $P, Q$ there is a line $g$ with $P \\mid g$ and $Q \\mid g$;\n2. if $P \\mid g$, $P \\mid h$, $Q \\mid g$ and $Q \\mid h$, then $P = Q$ or $g = h$;\n3. if $a, b, c \\mid P$ for a point $P$, then $abc \\in S$;\n4. if $a, b, c \\mid g$ for a line $g$, then $abc \\in S$;\n5. there are lines $g, h, j$ with $g \\mid h$ such that none of $j \\mid g$, $j \\mid h$, $j \\mid gh$ holds.\n\n## Remarks\n\nClauses 3 and 4 are the three reflections theorems.\nThe product of the reflections in three lines through a common point, or in three lines perpendicular to a common line, is again a reflection.\nClause 5 excludes degenerate cases.\n\nExample: let $G$ be the group of isometries of the Euclidean plane generated by the [[reflections|06BRDF3F0E]] in lines, and $S$ the set of these reflections.\nTwo reflections compose to an involution exactly when their lines are perpendicular, and then the product is the half-turn about the intersection point.\nA half-turn about $P$ times the reflection in $g$ is an involution exactly when $P$ lies on $g$.\nThe product of reflections in three concurrent lines, or in three parallel lines, is a reflection, which gives clauses 3 and 4.\n\nNon-example: let $G$ be cyclic of order $2$ and $S$ its non-identity element $s$.\nThen $s s = 1$ is not an involution, so there are no points and clause 5 fails.\n\nMetric planes give a group-theoretic form of plane [[absolute geometry|VKFPCSRNDH]].\nThe real hyperbolic plane and the real elliptic plane give further examples, with the same construction from their reflections.\n",
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