This paper considers the design of finite impulse response (FIR) filters with equation constraints in the frequency domain and proposes a design algorithm. The design problem is formulated as Chebyshev approximation subject to equation constraints. The optimal filter is proved to have a characteristic property that there exists a set of frequencies including extremal frequencies where the weighted error takes a maximum value, and constraint frequencies where the response takes a given magnitude. According to this property, an orthogonal projection-based Remez exchange algorithm is proposed to solve the constrained Chebyshev design problem. As its application, the algorithm is applied to the design of FIR lowpass filters and notch filters. Simulation examples demonstrate the effectiveness and good performance of the proposed algorithm.