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< APB, < BPC, < CPA right angles in space, given triangle ABC

Source: Austrian-Polish 1992

May 25, 2019
geometry3D geometrysolid geometrygeometric inequalityright angle

Problem Statement

Consider triangles ABCABC in space. (a) What condition must the angles ∠A,∠B,∠C\angle A, \angle B , \angle C of △ABC\triangle ABC fulfill in order that there is a point PP in space such that ∠APB,∠BPC,∠CPA\angle APB, \angle BPC, \angle CPA are right angles? (b) Let dd be the longest of the edges PA,PB,PCPA,PB,PC and let hh be the longest altitude of △ABC\triangle ABC. Show that 136h≤d≤h\frac{1}{3}\sqrt6 h \le d \le h.
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