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Rule of Thumb or Proveable?

Consider this page from a rare geometry text:

On the left-hand page, you will note the following rule in the form of Problem VII, along with several examples of its use:

Problem VII: The base and perpendicular of any plane triangle being given, to find the side of its inscribed square.

Rule: Divide the product of the base and perpendicular by their sums, and the quotient will be the side of the inscribed square.

My Question #1: Why can you assume that only one inscribed square is possible...or what other assumptions are being made?

My Question #2: Can you prove that this Rule works?

Solution:

Source: W. Vodges's Elementary Treatise on Mensuration and Practical Geometry, 1854, p. 74