As the title suggests, the half-angle formulas are central. Even more central than the law of cosines, which is nothing more than the half-angle formulas in disguise. Virtually, every metric relationship characterizing the triangle can be derived from the triangular half-angle formulas. The mind map below can give you an idea of the pivotal role that the half-angle formulas play in relation to the other most important metric relationships in classical geometry. I have also written an essay titled "The Theoretical Importance of Half-Angle Formulas" where you can see the details of the proofs, as well as the new generalizations I have managed to derive from this novel approach.
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Identifying the central theorems of an area is important because it helps streamline the process of understanding it. By knowing the basic principles, you can, with logic and a bit of ingenuity, understand (by proving) the rest of the discipline without having to memorize as much.
