Proposed Problem
See complete Problem 255 at:
gogeometry.com/problem/p255_triangle_centroid_squares_point_vertex.htm
Level: High School, SAT Prep, College geometry
Post your solutions or ideas in the comments.
Tuesday, February 24, 2009
Problem 255: Triangle, Centroid, Vertices, Interior/Exterior Point, Distances, Squares
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Let A be (0,0), B:(p,q) and C:(b,0). Hence, G, the centroid is ((p+b)/3,q/3). E is any point (x,y) in the Cartesian plane. Now LHS = a^2 +b^2 +c^2=p^2+q^2+b^2+(p-b)^2+q^2=2p^2+2q^2+2b^2-2pb ---(1). While, RHS = 3[x^2+y^2+(x-p)^2+(y-q)^2+(x-b)^2+ y^2-3[x^2+(p+b)^2/9-2x(p+b)/3+y^2+q^/9-2qy/3]]. On simplification most of the terms cancel out leaving RHS=2p^2+2q^2+2b^2-2pb = LHS by equation (1). Hence etc.
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