At its core ח־ר־ה means to burn, to flare up. The Bible says חרה אף, literally "the nose burned", for someone whose anger ignites. From that heat the root branches in two directions: the old wrath sense (חָרוֹן, burning indignation), and a modern twist where the same fire becomes competition. To compete (לְהִתְחָרוֹת) is to heat up against a rival. One image, anger as fire, feeds both the courtroom and the racetrack. Every word here 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 here is biblical and rarely heard in speech; modern Hebrew keeps the heat alive almost entirely through the hitpa'el לְהִתְחָרוֹת (to compete). The hif'il sense "to concur" is a separate, marginal usage. No pi'el, pu'al, nif'al or huf'al are lexicalized for this root.
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.