This paper proposes an algorithm for estimating the location of wireless access points (APs) in indoor environments to realize smartphone positioning based on Wi-Fi without pre-constructing a database. The proposed method is designed to overcome the main problem of existing positioning methods requiring the advance construction of a database with coordinates or precise AP location measurements. The proposed algorithm constructs a local coordinate system with the first four APs that are activated in turn, and estimates the AP installation location using Wi-Fi round-trip time (RTT) lateration and the ranging results between the APs. The effectiveness of the proposed algorithm is confirmed by conducting experiments in a real indoor environment consisting of two rooms of different sizes to evaluate the positioning performance of the algorithm. The experimental results showed the proposed algorithm using Wi-Fi RTT lateration delivers high smartphone positioning performance without a pre-constructed database or precise AP location measurements.