# A line segment joins the middle points of each diagonal of a quadrilateral with known sides. What is sum of the squares of the diagonals?

$$|DF|=|FB|$$

$$|AE|=|EC|$$

What is $$|AC|^2 + |BD|^2= \ ?$$

I asked one of my classmates for the solution, he said:

$$|AC|^2+|BD|^2+4 \cdot3^2 = 7^2 +4^2 +5^2+6^2$$

He didn't tell me how he obtained it, either because he wouldn't want to tell me (I remotely know him) or he just memorized the formula.

Provided that this is a high school geometry problem, can you show me how above equation is obtained in simple terms?

There is identity which states that for every quadrilateral $$ABCD$$ we have: $$AB^2+BC^2+CD^2+DA^2 = AC^2+BD^2+4EF^2$$

Proof: Remember that $$XY^2 = \vec{XY}^2 = (Y-X)^2 = Y^2-2X\cdot Y+X^2$$

Also remember also that $$E = {1\over 2}(A+C)$$ and $$F = {1\over 2}(B+D)$$.

So the left side is $$2A^2+2B^2+2C^2+2D^2-2A\cdot B-2B\cdot C-2C\cdot D-2D\cdot A$$

and the right side is $$A^2+C^2-2A\cdot C +B^2+D^2-2B\cdot D +(B+D-A-C)^2$$

Now it should not be difficult to see they are the same...

So use it and you are done.