At noon, all three hands are at $0$ degrees from each other.

Let's say the Moment of Truth is after the hour hand has moved around $x$ degrees. In that time, the minute hand moves $12x$ degrees and the second hand moves $720x$ degrees. So we want $x, 12x\pmod{360},$ $720x \pmod {360}$ to be separated by $120$ from each other.

From the minute and second hands we know $720x-12x\pm120$ is a multiple of $360$, which means $59x\pm10$ is a multiple of $30$. From the minute and hour hands we know $11x\pm120$ is a multiple of $360$. So $11x$ and $59x$ are both integers, which means $x$ is an integer (since 11 and 59 are coprime). But then $720x$ is a multiple of $360$, which means $x$ must be $120$ or $240 \pmod {360}$ and so $12x$ is another multiple of $360$ contradiction. So

>! there is no solution!