Warm up problem: Cartesian Right Triangle 2 (harder)

Geometry Level 4

here is a link to the easier one

Given point A=(1,6) and B=(4,12) and line D=3x+7, a point C is placed on line D such that ABC\triangle ABC is a right triangle. If the sum of all possible x-coordinates of c can be represented bypq\dfrac{p}{q}. Find p+q.

Tip: the right angle doesn't necessarily occur at A or B.

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