This article presents one human proof and three machine proofs of a challenging geometric theorem that gives a beautiful representation of the area of an arbitrary quadrilateral in terms of its four sides and four internal angles. These proofs demonstrate the power, capability and features of automated deduction methods and tools, which reduce qualitative difficulty to quantitative complexity versus traditional methods with individual ingenious ideas, for mathematical theorem proving. The present case study not only results in four probably new proofs of a hard theorem but also contributes to understanding the significance of developing effective algorithms and software tools for automated theorem proving in mathematics using advanced computing technology.