If the sum of squares of one set of opposite sides of a quadrilateral is equal to the sum of squares of another set of opposite sides, If the sum of squares of one set of opposite sides of a quadrilateral is equal to the sum of squares of another set of opposite sides, what is the relationship between its diagonals? Why?

If the sum of squares of one set of opposite sides of a quadrilateral is equal to the sum of squares of another set of opposite sides, If the sum of squares of one set of opposite sides of a quadrilateral is equal to the sum of squares of another set of opposite sides, what is the relationship between its diagonals? Why?

Its diagonal line seems to be vertical proof: make be ⊥ AC to e, DF ⊥ AC to F, then: ab ^ 2 = be ^ 2 + AE ^ 2CD ^ 2 = DF ^ 2 + CF ^ 2ad ^ 2 = DF ^ 2 + AF ^ 2BC ^ 2 = be ^ 2 + CE ^ 2, because AB ^ 2 + CD ^ 2 = ad ^ 2 + BC ^ 2, so AE ^ 2 + CF ^ 2 = AF ^ 2 + CE ^ 2, so e and F must coincide, that is: B, e, F and D are collinear, so BD ⊥ AC