Problem of the Week Problem E Now I Know My ABCs

In triangle $$ABC$$, point $$P$$ lies on $$AB$$, point $$Q$$ lies on $$BC$$, and point $$R$$ lies on $$AC$$ such that $$AQ$$, $$BR$$, and $$CP$$ are altitudes with lengths $$21$$ cm, $$24$$ cm, and $$56$$ cm, respectively.

Determine the measure, in degrees, of $$\angle ABC$$, and the lengths, in centimetres, of $$AB$$, $$BC$$, and $$CA$$.

Note the diagram is not drawn to scale.