Answer
See explanations.
Work Step by Step
Step 1. As the temperature around the Earth's equator is a continuous function, we can draw a curve as shown in the figure, where the starting point is any point on the equator and the endpoint is the same point with a total distance of $2\pi R$ where $R$ is the radius of the Earth.
Step 2. We need to show that if we draw horizontal lines (equal temperatures) across the curve, we can find a distance $D$ between the intersection points to be equal to $\pi R$ (diametrically opposite).
Step 3. The largest distance $D$ is $2\pi R\gt\pi R$, and the smallest is $0$. As this distance $D$ is a continuous function when we move the horizontal line across different temperatures, there must exist a temperature where $D=\pi R$ (The Intermediate Value Theorem).
Step 4. Thus, there is always a pair of antipodal (diametrically opposite) points on Earth’s equator where the temperatures are the same.
