2001 AIME I Problems/Problem 12

Problem

A sphere is inscribed in the tetrahedron whose vertices are and The radius of the sphere is where and are relatively prime positive integers. Find

Solution

import three; pointpen = black; pathpen = black+linewidth(0.7); currentprojection = perspective(-2,9,4);
triple A = (6,0,0), B = (0,4,0), C = (0,0,2), D = (0,0,0);
triple E = (2/3,0,0), F = (0,2/3,0), G = (0,0,2/3), L = (0,2/3,2/3), M = (2/3,0,2/3), N = (2/3,2/3,0);
triple I = (2/3,2/3,2/3);
triple J = (6/7,20/21,26/21);
draw(C--A--D--C--B--D--B--A--C)
draw(L--F--N--E--M--G--L--I--M--I--N--I--J);
label("$I$",I,W);
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,W*-1);
label("$D$",D,W*-1); (Error making remote request. Unknown error_msg)

The center of the insphere must be located at where is the sphere's radius. must also be a distance from the plane

The signed distance between a plane and a point can be calculated as , where G is any point on the plane, and P is a vector perpendicular to ABC.

A vector perpendicular to plane can be found as

Thus where the negative comes from the fact that we want to be in the opposite direction of

Finally

See also

• <url>viewtopic.php?p=384205#384205 Discussion on AoPS</url>

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