CBSE Class 11-science Answered

Let the vertices are A, B, and C, the side lengths opposite the vertices are a, b, and c and the medians are m_a, m_b, and m_c.

let, m_{a =} m_b 

Now, Apollonius’s theorem says

b²+c²=2(a/2)²+2m_a² and

c²+a²=2(b/2)²+2m_b².

Doubling both equations, and moving the squared median lengths to one side of each equation,

4m_a²=2b²+2c²−a², and

4m_b²=2c²+2a²−b².

Setting these squared median lengths equal to one another and

cancelling terms, we get

3b²=3a²

So the triangle is isosceles.