Solving plane geometry using vectors and conditions on angles

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I can use simple vectors to work out Euclidean properties of plane figures such as the location of the intersection of the bisectors of the sides of triangles (the centroid). Is it possible to use simple vectors to locate the intersection of the angle bisectors of the three vertices of the triangle (the incentre)? I haven't found this, but maybe I don't know where to look. Then an extension to locate points of intersection of angles divided in certain ratios?

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A useful property to note is that the incentre is the unique point with equal distances from all sides. Try using the projection formula to find the distances, and solve for the point equidistant from all sides.