To prove the identity: AB^2+AC^2=2BM^2+2AM^2, we have to use the law of cosines.

For the triangle AMB,

c^2 = x^2 + y^2 - 2*x*y*cos (AMB)...(1)

For the triangle AMC

b^2 = x^2 + y^2 - 2*x*y*cos (AMC)

as cos (AMC) = cos( 180 - AMB) = -cos (AMB), we get

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To prove the identity: AB^2+AC^2=2BM^2+2AM^2, we have to use the law of cosines.

For the triangle AMB,

c^2 = x^2 + y^2 - 2*x*y*cos (AMB)...(1)

For the triangle AMC

b^2 = x^2 + y^2 - 2*x*y*cos (AMC)

as cos (AMC) = cos( 180 - AMB) = -cos (AMB), we get

b^2 = x^2 + y^2 + 2*x*y*cos (AMB...(2)

Adding (1) and (2)

c^2 + b^2 = 2x^2 + 2y^2

c = AB , b = AC, x = BM, y = AM

=> AB^2 + AC^2 = 2BM^2 + 2AM^2

Therefore we prove **AB^2 + AC^2 = 2BM^2 + 2AM^2**