We all know Pythagorean Theorem: the legs a, b and a hypotenuse of any right triangle satisfy an equation:
The commonly known generalization of that theorem is Law of Cosines:
There is another generalization and it involves use of Orthocentre. Let be an Orthocentre of , let denote feet of altitudes from respectively.
When we use that as a substitution in the Law of Cosines we obtain: or
Since we conclude that quadrilateral is cyclic. Using power of point we can state that: and
Susbstituting it in the previous equation we get
In a special case when then and it becomes regular Pythagorean Theorem.
What is interesting is that if we create a right triangle on the other side of , with legs and hypotenuse , and repeat for sides we will get a net of a orthogonal tetrahedron with apex .
Orthogonal projection of apex onto is orthocentre . Projection of is etc.
It is commonly known that .
It can be easily shown when exploring tetrahedron that this constant is .
I currently do not have time to fill out all the details but I trust anybody who knows basics of geometry can fill in the blanks. I have come into realization of these facts which amuse me some years ago when solving brilliant Trirectangular Corner Locus and eventually I made an effort to write this as imperfect as it is. I thought it is better than not doing it at all. Writing, even notes or solutions can always be perfected, but fear of imperfection often means not doing it at all.