Three rods of equal length l are joined to form an equilateral triangle PQR. O is the midpoint of PQ. Distance OR remains same for small change in temperature. Coefficient of linear expansion for PR and RQ is same i.e. $\alpha_2$ but that for PQ is $\alpha_1$ . Then
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$(OR)^2 = (PR)^2 - (PO)^2 = 1^2 - \left(\frac{1}{2}\right)^2$
$= \left[\Pi(1 + \alpha_2 t)\right]^2 - \left[\frac{1}{2}(1 + \alpha_1 t)\right]^2$
$l^{2} - \frac{l^{2}}{4} = l^{2} \left( 1 + \alpha_{2}^{2} t^{2} + 2 \alpha_{2} t \right) - \frac{l^{2}}{4} \left( 1 + \alpha_{1}^{2} t^{2} + 2 \alpha_{1} t \right)$
Neglecting $\alpha_{2}^{2} t^{2}$ and $\alpha_{1}^{2} t^{2}$
$0 = l^{2}(2\alpha_{2}t) - \frac{l^{2}}{4}(2\alpha_{1}t) \Rightarrow 2\alpha_{2} = \frac{2\alpha_{1}}{4} \Rightarrow, \alpha_{1} = 4\alpha_{2}$
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