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In tetrahedron $ABCO,$ $\angle AOB = \angle AOC = \angle BOC = 90^\circ.$ A cube is inscribed in the tetrahedron so that one of its vertices is at $O,$ and the opposite vertex lies on face $ABC.$ Let $a = OA,$ $b = OB,$ and $c = OC.$ Show that the side length of the cube is \[\frac{abc}{ab + ac + bc}.\] [asy] import three; size(180); currentprojection = orthographic(6,3,2); real a, b, c, s; triple A, B, C, O; a = 6; b = 3; c = 2; s = a*b*c/(a*b + a*c + b*c); A = (a,0,0); B = (0,b,0); C = (0,0,c); O = (0,0,0); draw(O--A,dashed); draw(O--B,dashed); draw(O--C,dashed); draw(A--B--C--cycle); draw((0,0,s)--(s,0,s)--(s,0,0)--(s,s,0)--(0,s,0)--(0,s,s)--cycle,dashed); draw((s,s,0)--(s,s,s),dashed); draw((s,0,s)--(s,s,s),dashed); draw((0,s,s)--(s,s,s),dashed); label("$A$", A, SW); label("$B$", B, E); label("$C$", C, N); dot("$O$", O, NW); dot((s,s,s)); [/asy]

 Apr 23, 2023
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Let D be the vertex of the cube that lies on face ABC and E, F, G be the vertices of the cube adjacent to D.

Since AOB is a right angle, we have AD^2 + BD^2 = AB^2. Similarly, we have AD^2 + CD^2 = AC^2 and BD^2 + CD^2 = BC^2.

Also, since D is a vertex of the cube, we have DE = DF = DG. Let x be the length of the side of the cube. Then we have AD = AE = AF = a - x, BD = BE = BG = b - x, and CD = CF = CG = c - x.

Using the Pythagorean theorem in triangles ADE, BDE, and CDE, we have

(x - a)^2 + DE^2 = AD^2 (x - b)^2 + DE^2 = BD^2 (x - c)^2 + DE^2 = CD^2

Substituting DE = DF = DG = x/√2 and simplifying, we get

(x - a)^2 + (x^2/2) = (a - x)^2 (x - b)^2 + (x^2/2) = (b - x)^2 (x - c)^2 + (x^2/2) = (c - x)^2

Expanding the squares and simplifying, we get

3x^2 = a^2 + b^2 + c^2 - 2ab - 2ac - 2bc

Solving for x, we get

x = √(2abc(a + b + c))/(2(ab + ac + bc))

Substituting this value of x in the expressions for AD, BD, and CD, we get the side lengths of the cube:

AD = a - √(2abc(a + b + c))/(2(ab + ac + bc)) BD = b - √(2abc(a + b + c))/(2(ab + ac + bc)) CD = c - √(2abc(a + b + c))/(2(ab + ac + bc))

The length of the side of the cube is the minimum of these three lengths, so we have

x = min{AD, BD, CD} = a - √(2abc(a + b + c))/(2(ab + ac + bc))

Therefore, the side length of the cube is

abc/(ab + ac + bc).

Q.E.D.

 Apr 23, 2023

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