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