This contribution in regards to the Harmonious Pythagorean Tetrahedra conjecture: PYTHAGOREAN THEOREM GENERALIZATION as CONSEQUENCE of THREE-DIMENSIONS
My great appreciation to Walter Trump for his assistance on this topic as the trees often obscure the forest. Equally the same to William Walkington for facilitating our interaction. Mr. Walkington’s post “Magic Triangular Pyramids” found on his blog “Magic Squares, Spheres and Tori” provided motivation to seek a clearer understand of the degenerate pyramidal limits which Mr. Trump so graciously was able to provide. Please visit William’s blog for some exceedingly thought provoking content.
The 3 construction cylinders intersect with the x-y-plane in 3 pairs of parallel straight lines.
The pyramid degenerates if 3 of these lines intersect in one point (D). See graphic.
Base right triangle: ABC with c = AB, b = AC, a = BC and a < b.
Coordinates: C(0, 0), A(b, 0), B(0, a)
Cylinder radius: 2c, 2b, 2a
F is a point on the orange line with BF ⊥ BA.
G is the intersection of the orange line with the y-axis.
The triangles BFG, BFH and FGH are similar to the triangle ABC.
BF = 2c
FH : BF = a : c ⇒ FH : 2c = a : c ⇒ FH = 2a ⇒ F lays on the left green line
BG : BF = c : b ⇒ BG : 2c = c : b ⇒ BG = 2c² / b
CG = BG – BC = 2c² / b – a ⇒ G( 0 , – 2c² / b + a )
Equation of the orange line through G: y = (– a/b) × x – 2c² / b + a
This line intersects with the green line x = 2a and the blue line y = –2b exactly if
–2b = (– a/b) × 2a – 2c² / b + a
–2b – a = – 2a²/ b – 2c² / b
2b + a = 2a²/ b + 2c² / b
2b² + ab = 2a² + 2c²
2b² + ab = 2a² + 2a² + 2b²
ab = 4a²
b = 4a
Thus the pyramid degenerates if b = 4a (or a = 4b).
