1. From the condition at $12:00$, it follows that the tracks intersect at an angle of $60^{\circ}$. Since they crossed the intersection after this, they are moving towards it. Since the cyclist spent twice as much time on this road, his speed is half the speed of the motorcyclist, i.e., 36 km/h.

If we start the time count $t$ from 13:00 (negative values of $t$ are possible), then at time $t$, the motorcyclist is at a distance of $|72 t|$ from the intersection point, and the cyclist is at a distance of $|36(t-1)|$. Therefore, by the cosine theorem:

$$
72^{2} t^{2}+36^{2}(t-1)^{2}-2 \cdot 72 \cdot 36|t||t-1| \cdot \frac{1}{2}=252^{2}
$$

or after simplification:

$$
2^{2} t^{2}+(t-1)^{2}-2|t||t-1|=7^{2}
$$

Obviously, values of $t$ between 0 and 1 do not fit (this follows from a simple estimate of the distance between them), therefore: $4 t^{2}+t^{2}-2 t+1-2 t^{2}+2 t=49$, i.e., $t^{2}=16$. Thus, the required moments of time are: 09:00 and 17:00.
Answer: 09:00 and $17:00$.