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  "body": "\n## Statement\n\nLet $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane and let $P$ be a point inside $ABC$ or on its boundary.\nLet $p$, $q$, $r$ be the distances from $P$ to the lines $BC$, $CA$, $AB$.\nThen\n$$PA + PB + PC \\ge 2(p + q + r).$$\nEquality holds if and only if $ABC$ is equilateral and $P$ is its centre.\n\n## Remarks\n\nWrite $a = BC$, $b = CA$ and $c = AB$.\nOne proof reflects $P$ in the bisector of the angle at $A$ and compares areas; this gives $a \\cdot PA \\ge b r + c q$.\nAdding the three such bounds and using $t + 1/t \\ge 2$ for $t \u003e 0$ gives the inequality.\nExample: let $ABC$ be equilateral with side $2$ and let $P$ be its centre.\nEach distance to a vertex is $2/\\sqrt 3$ and each distance to a side is $1/\\sqrt 3$, so both sides equal $2\\sqrt 3$.\nThe hypothesis that $P$ lies in the closed triangle is needed.\nFor some points far outside the triangle, the distances to the side lines are large compared to the distances to the vertices, and the inequality fails.\nErdős posed the inequality as a problem, and Mordell and Barrow gave the first proof.\n\n## Explanations\n\n### L1 human\n\nStand at a point inside a triangle.\nAdd up your distances to the three corners, then add up your distances to the three sides.\nThe first sum is always at least twice the second.\nOnly the centre of an equilateral triangle makes the two sides equal.\n\n### L2 human\n\nThe inequality is a classic of olympiad geometry and a model for many inequalities that compare vertex distances with side distances.\nIts proofs use [[Euclidean distance|0NPPGHQNXQ]] and area arguments, or the law of cosines combined with the inequality between arithmetic and geometric means.\n",
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