Vector Addition?

It follows from the solution of the time-invariant two-dimensional heat equation $\nabla^2 u(x,y)=0$ that, temperature at any point in the steady-state is the average of the temperatures of the neighbouring points. Thus the temperature at the junction is $T=\frac{1}{4}\big(100+100+50+70)^\circ C=80^\circ C$

Assuming the value of junction as $\theta$ and applying Kirchoff's Junction rule for heat currents, we get

100 - $\theta$ + $\theta$ = $\theta$ - 70 + $\theta$ - 50 { heat entering the junction from the two rods at higher temperatures is equal to the heat exiting the junction to the rods at lower temperatures)

On solving, we get $\theta$ = 80