Trirectangular Vertex

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.

Click HERE for the Graphing Calculator file (if wanted).

Here is an image of the trirectangular vertex drawn with GSP.


Click HERE for 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