Hi there. I'm happy to contribute to this.

It seems to me, the information given is not a complete question, because there are a number of possibilities for the solutions. A = (1/3, 0), and D = (1/2, 0), so AD = 1/6.

Given that AC = BD, the only possible orders are:
(I) B, A, D, C
(II) A, B, C, D
(III) C, A, D, B
(IV) A, C, B, D

Case I: the order on the axis is B, A, D, C

Then, since AC = DB, we know that AB = CD. Call that distance x. AB = CD = x.

BC = 5x = AB + AD + DC = x + 1/6 + x

3x = 1/6 --> x = 1/18

Thus, B = (5/18, 0), and C = (10/18, 0). That's one solution.

Case II: the order on the axis is A, B, C, D

Call since AC = DB, we know that AB = CD. Call AB = CD = x

AD = AB + BC + CD

1/6 = x + 5x + x = 7x

x = 1/42

Then, B = (15/42, 0) and C = (20/42, 0)

Case III: C, A, D, B and Case IV: A, C, B, D are not possible, because BC must be 5x bigger than AB.

Those are all the possible cases, but we are left with two scenarios, one in which C = (10/18, 0) = (5/9, 0), and the other in which C = (20/42, 0) = (10/21, 0). If this is Problem Solving question, we would need more information to narrow things down to a definitive answer. Either that, or we could answer a question along the lines "which of the following could be a possible value of C?"

I hope this helps. Does it all make sense? Please let me know if you have any additional questions.

Mike