Given a trirectangular vertex. (0,0, 0) (p,0,0), (0, q, 0), (0,0,m) as coordinate of the vertices.

To illustrate, here is an image of a trirectangular vertex created by Graphing Calculator 4.0. The equations of the three mutually perpendicular planes are

x = 0

y = 0

z = 0

with the ranges set from 0 to 4 on each axis.

The equation for the slant plane is

px + qy + mz = 4

(The choice of 4 for the range and the constant is onlyh for convenience. We could have chosen any constant.)

Transparency has been enabled on each plane.

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HEREfor the Graphing Calculator file (if wanted).Here is an image of the trirectangular vertex drawn with GSP.

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HEREfor the GSP file (if wanted).

Let A be the area in the xy plane, B the area in the yz plane, C be the are in the xz plane.

Let D be the area of the slant plane face.

1. Express the area D in terms of A, B, and C.

Note: This is known as de Gua's Theorem

2. Express the volume V in terms of A, B, and C