Dancing Rods

Geometry Level 5

Let two Rods \(AB\) of length \(10\) and \(CD\) of length \(20\) are sliding on smooth standard co-ordinate axis such that their ends are always con-cyclic. If the locus of centre of that circle which pass through all four points \(A,B,C,D\) is expressed as:

\[ (ax-by)^2 + (bx-ay)^2 = c^2 \]

For positive integers \(a,b,c\). What is the minimum value of \(a+b+c\)

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