The triplets - 5, 12, 13

Geometry Level 4

Let \( A(5,12) \), \( B(-13\cos { \theta } ,13\sin { \theta } ) \) and \( C(13\sin { \theta } ,-13\cos { \theta } ) \) are angular points of \( \triangle ABC \) where \( \theta \in \Re \). The locus of the orthocentre of the \( \triangle ABC \) is \( ax+by+c=0 \) where \( a,b,c \) are integers. Find the minimum positive value of \( a+b+c \).

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