Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
Prove a quadrilateral is a rhombus using coordinates
Problem
Using the inclusive definition of a rhombus, verify that the ordered vertices \(A(0,2)\), \(B(2,0)\), \(C(4,2)\), \(D(2,4)\) form rhombus \(ABCD\). Give \(\text{all}~\text{four}~\text{squared}~\text{side}~\text{lengths}\), compare them, and state the conclusion.
Use the defining side condition and check all four cyclic edges, not just one opposite pair. Squared distances are the efficient invariant here: if the four positive squared lengths agree, then the actual lengths agree as well, so no separate square-root calculation is needed to support the classification.
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