Final answer:
The question seeks a proof that a given quadrilateral is a rhombus but provides insufficient or incorrect details for an adequate response.
Step-by-step explanation:
The question asks us to prove that a quadrilateral EFGH is a rhombus, given certain properties about its sides and angles. A rhombus is defined as a quadrilateral with all four sides of equal length. However, the information provided appears to be mixed with other unrelated concepts and details about moon measurements, vectors, and the Pythagorean Theorem that do not pertain to the geometry of quadrilateral EFGH. To prove that EFGH is a rhombus, we would normally look for details that demonstrate all sides are equal in length and that opposite angles are equal. The description of congruent segments EC and CG, as well as HC and CF, and the right angle at HF could be a starting point, but the details given seem to be insufficient or incorrect for a complete proof without an accurate diagram or additional relevant information about EFGH.