math
posted by POD .
in a triangle BCA there are three medians b,c,a. Prove that
3(A^2+B^2+C^2)=4(a^2+b^2+c^2)
(sides) = (medians)

The standard notation for the length of medians:
a=(1/2)*sqrt(2C^2+2B^2A^2)
b=(1/2)*sqrt(2C^2+2A^2B^2)
c=(1/2)*sqrt(2A^2+2B^2C^2)
square both sides:
a^2=(1/4)*(2C^2+2B^2A^2)
b^2=(1/4)*(2C^2+2A^2B^2)
c^2=(1/4)*(2A^2+2B^2C^2)
multiply 4 on both sides:
4a^2=(2C^2+2B^2A^2)
4b^2=(2C^2+2A^2B^2)
4c^2=(2A^2+2B^2C^2)
add them all vertically:
you get:
4(a^2+b^2+c^2)=3(A^2+B^2+C^2)
Respond to this Question
Similar Questions

Math
A triangle has vertices X(0,0), Y(4,4), and Z (8,4). a) Write an equation for each of the three medians b) Recall that the centroid of a triangle is the point of intersection of the medians of the triangle. Use the equations from … 
geometry
Write a two column proof. Given CE = CA, EB and AD are medians, Prove: EB = AD for a triangle 
geometry
If two medians of a triangle are equal, prove that the triangle formed by a segment of each median and the third side is an isosceles triangle. 
geometry question proof needed
Medians AX and BY of Triangle ABC are perpendicular at point G. Prove that AB=CG. 
Geometry
Given a triangle ABC with A(6b,6c) B(0,0) and C (6a,0), prove that the medians of the triangle are concurrent at a point that is two thirds of the way from any vertex to the midpoint of the opposite side. I'm not sure how to prove … 
Math
12. In a triangle ABC, AC = 36, BC = 48, and the medians BD and AE to sides AC and BC, respectively, are perpendicular. Find AB. 
Geometry
How do you find the lengths of the sides of an isosceles triangle with three medians? 
Geometry
divide a triangle by the three medians. Prove that the area of those 6 sections are congruent. 
Math
In triangle ABC, the medians AD,BE, and CF concur at the centroid G. (a) Prove that AD < (AB + AC)/2. (b) Let P=AB+AC+BC be the perimeter of triangle ABC. Prove that 3P/4 < AD + BE + CF < P. 
Math any help would be greatly appreciated
In triangle ABC, the medians AD,BE, and CF concur at the centroid G. (a) Prove that AD < (AB + AC)/2. (b) Let P=AB+AC+BC be the perimeter of triangle ABC. Prove that 3P/4 < AD + BE + CF < P.