<Solid-2025-267>, hint

Let A" the feet of AA" to triangle ABC, A' the feet of  AA' to BCD,

By three perpendiculars theorem AA'A" is a right triangle .

Let A"' be the foot of dropped from A onto dihedral plane of <AA"A'=X

since <AA"A"' =1/2 (<AA"A'=X), A"A"'=AA" (=h)cos X/2.

Let A"" be  feet of altitude A"'A"". A"'A""=A"A"' sin X/2=h cosX/2 sinX/2=1/2 h sin X.

Hence A"',B"',C"'  are all h/2 high from BCD .