To determine the relationship between resistances (r1, r2, and r3) when the power consumption is equal, we need to analyze the circuit configuration and use the relevant formulas for power and resistance.



The given circuit configuration consists of r1, r2, and r3. R2 and R3 are in parallel to each other, and their equivalent resistance is in series with r1.

```
------- R2 -------
| |
r1 R3
| |
------- R2 -------
```



Power consumption can be calculated using the formula:

P = (V^2) / R

where P is the power consumed, V is the voltage across the resistor, and R is the resistance.

Since power consumption is the same in all three resistors (r1, r2, and r3), we can equate the power equations for each resistor:

P1 = P2 = P3

(V1^2) / r1 = (V2^2) / r2 = (V3^2) / r3



To find the relationship between r1, r2, and r3, we can compare the power equations for each resistor.

First, let's consider the relationship between r2 and r3. Since r2 and r3 are in parallel, their equivalent resistance can be calculated using the formula:

1/Req = 1/r2 + 1/r3

Since the power consumption is the same for r2 and r3, we can equate their power equations:

(V2^2) / r2 = (V3^2) / r3

From these two equations, we can eliminate V2 and V3 by equating them:

(V1^2) / r1 = (V2^2) / r2 = (V3^2) / r3

Cross-multiplying these equations, we get:

(V1^2) * r2 = (V2^2) * r1
(V1^2) * r3 = (V3^2) * r1

Dividing the equations, we get:

r2 / r3 = (V2^2) / (V3^2)

From the equation 1/Req = 1/r2 + 1/r3, we can substitute r2 / r3 with 1/Req:

1/Req = (V2^2) / (V3^2)

Since the power consumption is the same for r2 and r3, the voltage ratio across them is the same. Therefore, (V2^2) / (V3^2) is equal to 1.

1/Req = 1

This means that r2 and r3 are equal to each other:

r2 = r3

Substituting this relationship back into the equation 1/Req = 1/r2 + 1/r3, we get:

1/Req = 1/r2 + 1/r2

Simplifying this equation, we find:

1