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本文仅讨论特殊的三棱锥(即四面体)顶点的射影位置与底面三角形的“五心”的位置关系。 命题1 在三棱锥中,若三条侧棱的长相等,则顶点在底面上的射影为底面三角形的外心。 证明(略)。 由此还可得推论. 推论:在三棱锥中,若侧棱与底面所成的角都相等,则顶点在底面上的射影为底面三角形的外心。 例1 有—三棱锥的高是h,侧棱与底面所成的角都是φ,底面是两个角分别为α和β的三角形,求它的体积(α、β都为锐角).
This article only discusses the positional relationship between the projective position of a particular trigonal pyramid (ie, tetrahedron) vertex and the “five heart” of the base triangle. Proposition 1 In a triangular pyramid, if the lengths of the three side edges are equal, then the projection of the vertex on the bottom surface is the outer center of the bottom triangle. Proof (omitted). It can also be inferred. Corollary: In the triangular pyramid, if the angle formed by the side edge and the bottom surface are equal, then the projection of the vertex on the bottom surface is the outer center of the bottom triangle. Example 1 has - the height of the triangular pyramid is h, the angle formed by the side edge and the bottom surface is φ, the bottom surface is a triangle with two angles α and β respectively, and its volume (α, β are all acute angles).