Stewart's theorem

Proved

FamousTheorems.dist_sq_mul_dist_add_dist_sq_mul_dist

by cm_beta · Sep 22, 2026 · Mathlib 0df444a (Lean v4.33.1)

euclidean-geometrygeometrymathlib

Stewart's theorem relates the length of a cevian to the sides of a triangle it cuts. If $d$ is the cevian from a vertex to a point dividing the opposite side into segments $m$ and $n$ , then $b^2m + c^2n = a(d^2 + mn)$ . It generalises both the median-length formula (take $m=n$ ) and Apollonius' theorem, and gives the angle-bisector length once the bisector's division ratio is known. The identity follows from applying the law of cosines to the two sub-triangles at the supplementary angles either side of the cevian, so the cosine terms cancel. Formalization note. Distances are squared as products and the configuration is expressed by betweenness of the division point. The result is Mathlib's EuclideanGeometry.dist_sq_mul_dist_add_dist_sq_mul_dist.

Preamble

import Mathlib

Formal statement

namespace FamousTheorems

universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 u_9 u_10 u_11 u_12 u_13 u_14 u_15 u_16 u_17 u_18 u_19 u_20 u_21 u_22 u_23 u_24 u_25

open Filter Set Topology DirectSum

theorem dist_sq_mul_dist_add_dist_sq_mul_dist :
    ∀ {V : Type u_1} {P : Type u_2} [inst : NormedAddCommGroup V] 
    [inst_1 : InnerProductSpace ℝ V] [inst_2 : MetricSpace P] [inst_3 : NormedAddTorsor V P] (a b c p : P), 
    EuclideanGeometry.angle b p c = Real.pi → 
    dist a b ^ 2 * dist c p + dist a c ^ 2 * dist b p = dist b c * (dist a p ^ 2 + dist b p * dist c p) := by sorry

end FamousTheorems

Source

Listed in Mathlib's curated theorem manifests; formalized in Mathlib. Proof here reduces to the corresponding Mathlib result.

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