Let the difference of temperature between angle θ and angle θ+π be f(θ). If f(θ)=0 when θ=0, we have already found two antipodal points which have a same temperature. If f(0) is not 0, then f(0)>0 f(π)<0 or f(0)<0 f(π)>0. The circle is a closed figure, so the graph which show the relationship between θ and f(θ) is continuous, no matter what the graph looks like. The change of θ is similar with a running race. When you are the NO.10 initially and your aim is NO.1, you need to exceed everyone in front of you. The first step is to exceed the one who is right in front of you and become NO.9. Repeat this process several times, you will be the NO.1. When we review the whole process, you have at least once to be in each position on the court, because you keep running during the race. As same as the graph, when you start at f(0) and try to reach f(π), you have to experience each point between f(0) and f(π) at least once. As we known, f(0) and f(π) have different sign, so 0 must between them. That is why there exist a number a in (0,π) such that f(a)=0.