This paper presents a new framework based on geometric algebra (GA) for solving and analysing three-phase balanced electrical circuits under sinusoidal and non-sinusoidal conditions. The proposed approach is an application of the geometric algebra power theory (GAPoT) to three-phase systems. Calculations are performed in a multi-dimensional Euclidean space where cross effects between voltage and current harmonics are taken into consideration. A definition of geometric apparent power for three-phase systems that complies with the energy conservation theorem is introduced. By using the proposed framework, the current can be easily decomposed into active- and non-active components for current compensation purposes. The paper includes detailed examples in which electrical circuits are solved and the results are analysed. This work is a first step towards a more advanced polyphase proposal with realistic cases, where unbalance and asymmetry is included.