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  "body": "\n## Statement\n\nLet $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane, and let $D$ be a point on the segment $BC$.\nWrite $a = BC$, $b = CA$, $c = AB$, $m = BD$, $n = DC$ and $d = AD$ for the [[distances|0NPPGHQNXQ]].\nThen\n$$b^2 m + c^2 n = a \\left( d^2 + m n \\right).$$\n\n## Remarks\n\nIf $D$ lies strictly between $B$ and $C$, the angles $\\angle ADB$ and $\\angle ADC$ add up to $\\pi$, so their cosines are opposite.\nApply the [[law of cosines|060ZVZBV3E]] to the triangles $ABD$ and $ADC$ at the vertex $D$.\nMultiply the first equation by $n$, the second by $m$, and add: the cosine terms cancel, and $m + n = a$ gives the formula.\nIf $D = B$ or $D = C$, the formula is direct.\n\nExample: if $D$ is the midpoint of $BC$, then $m = n = a/2$, and the formula becomes $d^2 = \\frac{b^2 + c^2}{2} - \\frac{a^2}{4}$.\nThis is [[Apollonius's theorem|0NAKHQ9XYY]] on the length of a [[median|0XW8HX8VEV]].\n\nFor a point $D$ on the line $BC$ outside the segment, the formula holds with signed lengths $m$ and $n$.\n\n## Explanations\n\n### L1 human\n\nDraw a segment from one corner of a triangle to any point of the opposite side.\nStewart's theorem gives the length of that segment from the three sides and the two pieces into which it cuts the opposite side.\nNo angles are needed.\n\n### L2 human\n\nStewart's theorem is the general form of several familiar length formulas.\nWith the midpoint it gives the length of a median.\nWith the foot of an angle bisector, located by the [[angle bisector theorem|0859R2V6ES]], it gives the length of the bisector.\n",
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