MathDB

8

Part of 1982 Austrian-Polish Competition

Problems(1)

d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD)>=3/2\sqrt2, r.tetrahedron

Source: Austrian-Polish 1982

5/25/2019
Let PP be a point inside a regular tetrahedron ABCD with edge length 11. Show that d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD)≥322d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD) \ge \frac{3}{2} \sqrt2 , with equality only when PP is the centroid of ABCDABCD. Here d(P,XY)d(P,XY) denotes the distance from point PP to line XYXY.
geometry3D geometrysolid geometrygeometric inequalitytetrahedron