Proposed Problem
See complete Problem 249 at:
gogeometry.com/problem/p249_triangle_sides_medians_squares.htm
Level: High School, SAT Prep, College geometry
Post your solutions or ideas in the comments.
Sunday, February 15, 2009
Problem 249. Triangle sides, medians, squares
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We apply median's theorem, Stewart's theorem and the properties of centroid. It's easy !
ReplyDeleteLet A:(0,0), B(q,r) and C:(p.0). Using distance formula, it's easy to show that:
ReplyDeletem(a)^2+m(b)^2+m(c)^2=(3/2)[p^2-pq+q^2+r^2)
while a^+b^2+c^2=(2)[p^2-pq+q^2+r^2)
This gives us:
(2/3)[m(a)^2+m(b)^2+m(c)^2]=(1/2)(a^+b^2+c^2)
or (a^+b^2+c^2)=(4/3)[m(a)^2+m(b)^2+m(c)^2]
Ajit: ajitathle@gmail.com