Tetrahedron inscribed and circumscribed ellipsoids volumes ratio

Geometry Level pending

Given an arbitrary tetrahedron, if $V_1$ is the maximum volume of an inscribed ellipsoid inside it (i.e. ellipsoid inside it touching its four faces), and $V_2$ is the minimum volume of a circumscribed ellipsoid outside it (i.e. ellipsoid passing through its four vertices), then $V_2 = k V_1$ , for a positive integer $k$ . Find $k$ .

The answer is 27.

1 solution

Hongqi Wang
Dec 6, 2020

I got the answer from regular tetrahedron: line mid point E of a side CD and two opposite vertexes A, B respectively then get a △ having 3 sides $\sqrt 3 : \sqrt 3 : 2$ . Obviously the orthocenter O of this △ is the center of tetrahedron. Then we can get that: the length from center O to vertex A is $OA = \frac {\sqrt 6}{4}$ , and from center O to face's center H is $OH = \frac {\sqrt 6}{12}$ .

Tetrahedron inscribed and circumscribed ellipsoids volumes ratio

The answer is 27.

1 solution

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