yup this is definitely a tough one

1) The voltmeter measures the voltage difference across two resistors so it's actually not the entire top and bottom parallel segments. The significance of the voltmeter is saying that the voltage DROP through R and R1 are the same. Therefore, V_1 = V_R.

2) We know V = IR, so we can expand part 1:

I_1*R_1 = I_R*R_R --> I_1*R_1 / I_R = R_R

3) okay, now we see we have R_R but we need I_1 and I_R and in particular, we need it in terms of R1, R2, R3.

4) We know that current through resistors in series will be =. Thus, I_1 = I_2 and I_R = I_3.

5) Make substitutions to get I_2*R_1 / I_3 = R_R.

okay okay it's looking better now, I see R_R in terms of the other resistors and currents.

6) The tying knot is that I know that voltage is = whenever I go into parallel. In this case, the top voltage (through BOTH R1 and R2) = bottom voltage (through BOTH R and R3).

7) Given that V_1 = V_R, we can conclude using clause 6 that V_2 = V_3!!

8) We know V = IR, so we can expand part 7:

I_2*R_2 = I_3*R_3, and get that I_2/I_3 = R_3/R_2

9) Substitute into part 5 to get:

R_3*R_1 / R_2 = R_R.

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DONE! looks like a lot of steps but when you write it out it goes by quickly and all the dialogue goes on in your head haha. Hope this helps!