If $A,B,C\in \mathbb{R}^2$ form a triangle and $P$ is a point in the interior then, if $N$ is where the perpendicular from $P$ to $AB$ meets the triangle, and similarly for $M$ and $L$, $$\overline{PA}+\overline{PB}+\overline{PC}\geq 2(\overline{PM}+\overline{PN}+\overline{PL}).$$

1∀ (A B C P L M N : EuclideanSpace ℝ (Fin 2)),2  AffineIndependent ℝ ![A, B, C] →3    P ∈ interior ((convexHull ℝ) {A, B, C}) →4      N ∈ affineSpan ℝ {A, B} →5        (affineSpan ℝ {P, N}).direction ⟂ (affineSpan ℝ {A, B}).direction →6          M ∈ affineSpan ℝ {B, C} →7            (affineSpan ℝ {P, M}).direction ⟂ (affineSpan ℝ {B, C}).direction →8              L ∈ affineSpan ℝ {C, A} →9                (affineSpan ℝ {P, L}).direction ⟂ (affineSpan ℝ {C, A}).direction →10                  dist P A + dist P B + dist P C ≥ 2 * (dist P M + dist P N + dist P L)