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  "body": "\n## Definition\n\nLet $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean space $\\mathbb{R}^n$, and let $\\cdot$ be the dot product. Let $H_A$ be the orthogonal projection of $A$ onto the [[line|0MP9BCF6DP]] $BC$:\n$$H_A = B + \\frac{(A - B) \\cdot (C - B)}{(C - B) \\cdot (C - B)} \\, (C - B).$$\nThe altitude of $ABC$ from $A$ is the segment from $A$ to $H_A$. The point $H_A$ is the foot of the altitude, and $h_a = \\|A - H_A\\|$ is its length.\nThe altitudes from $B$ and from $C$ are defined in the same way.\n\n## Remarks\n\nThe formula uses the [[vector projection|0JW13B48YP]] of $A - B$ onto $C - B$. The length $h_a$ is the [[distance from $A$ to the line $BC$|01AAYNSBN6]].\nThe word altitude also names the whole line through $A$ and $H_A$, or the length $h_a$. Mathlib's `Affine.Simplex.altitude` is the line.\nWith $a = \\|C - B\\|$, the area of the triangle is $\\frac{1}{2} a h_a$.\nLet the right angle of a right triangle be at $C$. Then the leg $AC$ is the altitude from $A$, and the leg $BC$ is the altitude from $B$.\n\n## Explanations\n\n### L1 human\n\nAn altitude is the height of a triangle, measured from one corner. Drop a plumb line from the corner to the opposite side, or to its extension. The foot of the altitude can be outside the triangle when the triangle has an obtuse angle.\n",
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