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 A Point and a Cube (Posted on 2004-07-16)
Before tackling this one, take a look at this one.
```     +-------------D
/|            /|
/ |           / |
/  |          /  |
/   |         /   |
C-------------+    |
|    |        |    |
|    B--------|----+
|   /         |   /
|  /          |  /
| /           | /
|/            |/
+-------------A
```
A, B, C and D are non-adjacent vertices of a cube. There is a point P in space such that PA=3, PB=5, PC=7, and PD=6. Find the distance from P to the other four vertices and find the length of the edge of the cube.
There are two answers, one with P outside the cube and one with P inside the cube.

 No Solution Yet Submitted by Brian Smith Rating: 4.0000 (5 votes)

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 Final Cut | Comment 20 of 23 |

The following is my final cut at the problem. Vertex labeling as follows:

A(s,0,0)  a = PA = 3
B(0,s,0)  b = PB = 5
C(0,0,s)  c = PC = 7
D(s,s,s)  d = PD
E(s,s,0)  e = PE
F(s,0,s)  f = PF
G(0,s,s)  g = PG           P(x,y,z)
O(0,0,0)  o = PO

The program variable \$i represents the variable i and \$i2 = i^2.

\$t2 represents the sqrt(8*(19-d^2)*(d*2-64))

The program was run for d = 5, 6, and 7. The values for x, y, and z
differ from Charlie's because we labeled the vertices differently.

for(\$d=5;\$d<8;\$d++) {

\$d2 = \$d*\$d;
\$t2 = sqrt(8*(19-\$d2)*(\$d2-64));
\$s2 = (\$d2+83-\$t2)/6;
\$s = sqrt(\$s2);
\$e = sqrt((\$d2-15)/2);
\$f = sqrt((\$d2+33)/2);
\$g = sqrt((\$d2+65)/2);
\$o = sqrt((83-\$d2)/2);
\$x = (2*\$s2-\$d2+65)/(4*\$s);
\$y = (2*\$s2-\$d2+33)/(4*\$s);
\$z = (2*\$s2-\$d2-15)/(4*\$s);
printf("d = %f e = %f f = %f g = %f o = %f\n",\$d,\$e,\$f,\$g,\$o);
printf("       s = %f x = %f y = %f z = %f\n",\$s,\$x,\$y,\$z);
\$s2 = \$s2+\$t2/3;
\$s = sqrt(\$s2);
\$x = (2*\$s2-\$d2+65)/(4*\$s);
\$y = (2*\$s2-\$d2+33)/(4*\$s);
\$z = (2*\$s2-\$d2-15)/(4*\$s);
printf("       s = %f x = %f y = %f z = %f\n\n",\$s,\$x,\$y,\$z);

}

d = 5.000000 e = 2.236068 f = 5.385165 g = 6.708204 o = 5.385165
s = 3.284646 x = 4.686791 y = 2.251217 z = -1.402146
s = 5.021066 x = 4.502142 y = 2.908855 z = 0.518924

d = 6.000000 e = 3.240370 f = 5.873670 g = 7.106335 o = 4.847680
s = 3.090065 x = 3.891261 y = 1.302319 z = -2.581095
s = 5.488002 x = 4.065065 y = 2.607339 z = 0.420751

d = 7.000000 e = 4.123106 f = 6.403124 g = 7.549834 o = 4.123106
s = 3.464102 x = 2.886751 y = 0.577350 z = -2.886751
s = 5.656854 x = 3.535534 y = 2.121320 z = 0.000000

 Posted by Bractals on 2004-07-22 11:31:24

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