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On tetrahedron ABCD and its altitudes

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October 3, 2010
geometry3D geometrytetrahedrongeometry proposed

Problem Statement

(a) Given a tetrahedron ABCDABCD and its four altitudes (i.e., lines through each vertex, perpendicular to the opposite face), assume that the altitude dropped from DD passes through the orthocenter H4H_4 of △ABC\triangle ABC. Prove that this altitude DH4DH_4 intersects all the other three altitudes.
(b) If we further know that a second altitude, say the one from vertex A to the face BCDBCD, also passes through the orthocenter H1H_1 of △BCD\triangle BCD, then prove that all four altitudes are concurrent and each one passes through the orthocenter of the respective triangle.
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