Whether Parallax be Valid as a Distance Measure

It would seem it is.

In Heliocentrism, as Earth is moving annually, stars show at apparently different angles. This is known as parallax. However, as with Tychonian orbits, one may presume one is dealing with an inversion of the geometry ... not.

If parallax phenomenon so called is really a proper movement, danced by angels, then it is not mechanically tied to annual movement of Sun.

The angel may be performing the "parallax" at same distance as presumed by Heliocentrics, in an annual orbit as great as that of the Sun, this would be an inversion of the geometry. But it could also be performing it at a much closer range to us within a much smaller orbit, proportionally to how much closer. Or, theoretically, but less probably, further away and in a larger annual orbit.

See Diagram (click image to see larger version): Notice how triangle to the left has a much smaller "base" at star than the one at the right has "at orbit of the earth around sun".

Yet, the angles are the same - it is just that with Geocentrism we have no known distance.

You know, in trigonometry, a triangle can be defined by six measures, namely three angles and three distances. To know all six, you to know need three of them, but one of these three has to be a known distance.

Geocentrism leaves annual orbit of Sun around Earth outside the triangle where parallax angle is the sharp angle, the triangle which involves the star at two positions.

But that is the inverse of "Earth's annual orbit around Sun" which is on the contrary inside the "parallax trigonometry triangle" of Heliocentrism. The one known distance according to Heliocentrism does not belong to the triangle in Geocentrism.