One idea runs through this whole family: holding firm. First it is a quality, being strong (חָזָק). Then it becomes an act: you make something strong (לְחַזֵּק), or you grip it and don't let go (לְהַחְזִיק). From that grip grow the abstract nouns: a legal presumption that holds, the power of a number, and the upkeep that keeps machines and buildings strong over time. Strength, grip and maintenance are the same hand seen at three moments. Every word below is verified against pealim, niqqud included.
The same root runs through the verb system. Each binyan is a voice: it bends the one idea into active, passive, causative or reciprocal. Tap to hear.
The pa'al "to be strong" survives mostly in biblical and set phrases (חֲזַק! "be strong!"); modern Hebrew prefers the adjective חָזָק for the state. The living verbs are the pi'el (make strong) and the hif'il, which shifts the sense from strength to holding, the grip that keeps something in place.
Same root letters, different noun moulds (mishkalim). The mould tells you the function: who does it, the act of doing it, or the thing produced.
Strip the prefixes, suffixes and vowels. Three consonants remain, and the meaning lives there.