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  "body": "\n## Definition\n\nLet $ABC$ be a triangle in the Euclidean plane $\\mathbb{R}^2$, with [[incentre|0M9MX8YW0A]] $I$ and inradius $r$. Let $d$ be the Euclidean distance.\nThe incircle of $ABC$ is the [[circle|060W5CBZP6]] with the centre $I$ and the radius $r$:\n$$\\{\\, X \\in \\mathbb{R}^2 \\;:\\; d(X, I) = r \\,\\}.$$\n\n## Remarks\n\nThe incircle touches each side of the triangle at exactly one point, the foot of the perpendicular from $I$ to that side.\nLet $T$ be the area of $ABC$ and $s$ half of its perimeter. Then $T = r s$, so $r = T / s$.\nAn excircle is a circle outside the triangle that touches one side and the extensions of the other two sides. Each triangle has three excircles.\n\n## Explanations\n\n### L1 human\n\nThe incircle is the biggest circle that fits inside a triangle. It touches each side once. Cut the triangle into three pieces from $I$ to the corners: each piece has the height $r$, and this gives $T = r s$.\n",
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