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  "body": "\n## Definition\n\nLet $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane and let $\\Gamma$ be its [[circumcircle|03VZJE6C8C]].\nLet $P$ be a point of $\\Gamma$.\nLet $D$, $E$ and $F$ be the feet of the perpendiculars from $P$ to the [[lines|0MP9BCF6DP]] $BC$, $CA$ and $AB$.\nThe points $D$, $E$ and $F$ are [[collinear|0AFNNTRTJE]], and the *Simson line* of $P$ with respect to $ABC$ is the line that contains them.\n\n## Remarks\n\nThe collinearity in the definition is the Simson–Wallace theorem.\nConversely, if the three feet from a point $P$ are collinear, then $P$ lies on $\\Gamma$.\n\nExample: let $P = A'$ be the point of $\\Gamma$ opposite to $A$.\nSince $AA'$ is a diameter, the angles $\\angle ABA'$ and $\\angle ACA'$ are right angles, so the feet on $AB$ and $CA$ are $B$ and $C$.\nThe foot on $BC$ lies on the line $BC$, so the Simson line of $A'$ is the line $BC$.\n\nNon-example: let $P$ be the [[circumcentre|0F98FZK14A]] $O$, which is not on $\\Gamma$.\nThe feet of the perpendiculars from $O$ are the [[midpoints|0S706JBZBG]] of the three sides.\nThese form the medial triangle, so they are not collinear, and no Simson line is defined.\n",
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