Find opposite vertices of a rhombus, given the other 2

Question

I am stuck with this problem. I posted an earlier problem with a square, where rotation with i of 90 degrees was possible. This one is a rhombus, how should I proceed?

Given ABCD is a rhombus with AC = 2BD, and b = 3 + i, d = 1 - 3i. Find a and c.

Thanks for your help.

Answer 1

Since you're given opposite vertices and information about the lengths of the diagonals, it is probably useful to know some things about specific quadrilaterals. Kites—quadrilaterals where two consecutive sides are congruent and the other two (consecutive) sides are congruent—have perpendicular diagonals. Parallelograms have diagonals that bisect each other (that intersect at their midpoints). Rhombi (rhombuses) are simultaneously kites and parallelograms, which gives you some information about the diagonals of the rhombus in your particular problem.

Answer 2

Hint: What is true about all the sides of a rhombus?

Answer 3

Hint: The points $A,C$ are the points of intersection between the diagonal $AC$ and the circle centered at the mid point of $BD$ with radius $|\frac{AC}{2}|$. How do you express this information in terms of Cartesian equations?