Final answer:
The points of concurrency for triangle ABC with given vertices can be found using geometric properties of right triangles and formulas for centroids and incenters.
Step-by-step explanation:
The question pertains to finding the points of concurrency in a triangle, which include the centroid, circumcenter, orthocenter, and incenter of a triangle. Since the given vertices A(1, 4), B(-3, 4), and C(1, 1) form a right triangle with the right angle at C, the circumcenter which is the midpoint of the hypotenuse will be at the midpoint of AB.
The orthocenter of a right triangle coincides with the vertex of the right angle, which is C(1, 1) in this case. The centroid can be found by averaging the x-coordinates and y-coordinates of the vertices, which results in the point (1/3*[-3+1+1], 1/3*[4+4+1]) = (-1/3, 3). The incenter can be calculated using the formula that combines the coordinates of the vertices weighted by the lengths of the opposite sides.