First Triumvirate Triumvir or tresvir: member of a college of three members. The expression is mostly used to describe the First Triumvirate (60 BCE; Pompey the Great, Crassus, and Julius Caesar) and Second Triumvirate (43 BCE; Marc Antony, Lepidus, and Octavian).
With 8 nights, I suggest you spend 3-4 in Tokyo, 1 in Hakone for ryokan, 3-4 in Kyoto. Hakone offers Mt. Fuji views on a clear day, and it sits right off the train line between Tokyo and Kyoto so it's an easy detour. While at Kyoto, spend a day at Nara.
Great hiking spot close to Tokyo. A few dozens of minutes from the heart of Tokyo is Takao-san, a mountain which Japanese people have revered for a very long time. They'll refer about it with terms such as hiking but for most of us, it's quite a leisurely stroll on a concrete trail, accessible to wheelchairs and with toilets at regular intervals.