Generalization of Pythagorean Theorem with Orthocentre

We all know Pythagorean Theorem: the legs a, b and a hypotenuse $c$ of any right triangle satisfy an equation: $a^2 + b^2 = c^2$

The commonly known generalization of that theorem is Law of Cosines:

$c^2=a^2 + b^2 - 2 a \times b \times cos(C)$

There is another generalization and it involves use of Orthocentre. Let $H$ be an Orthocentre of $\triangle ABC$, let $D,E,F$ denote feet of altitudes from $A,B,C$ respectively.

$cos(C) = \dfrac{CE}{a} = \dfrac{CD}{b}$

When we use that as a substitution in the Law of Cosines we obtain: $c^2 = a(a-CD) + b(b-CE) = a \times BD + b \times AE$ or $AB^2 = AC \times AE+ BC \times BD$

Since $\angle ADE = 90, \angle BEC = 90$ we conclude that quadrilateral $CEHD$ is cyclic. Using power of point we can state that: $AE \times AC = AH \times AD$ and $BD \times BC = BH \times BE$

Susbstituting it in the previous equation we get

$AB^2 = AH \times AD+ BH \times BE$

In a special case when $\angle C = 90$ then $H=C=D=E$ and it becomes regular Pythagorean Theorem.

What is interesting is that if we create a right triangle on the other side of $AB$, with legs $AP=\sqrt{ AH \times AD}, BP= \sqrt{ BH \times BE}$ and hypotenuse $AB$, and repeat for sides $BC, AC$ we will get a net of a orthogonal tetrahedron with apex $P$.

Orthogonal projection of apex $P$ onto $\triangle ABC$ is orthocentre $H$. Projection of $AP$ is $AH$ etc.

It is commonly known that $AH \times HD= BH \times HE = CH \times HF$.

It can be easily shown when exploring tetrahedron $ABCP$ that this constant is $HP^2$.

$HP^2 = AH \times HD= BH \times HE = CH \times HF$

PS

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.