Geoville County wants to build a hospital that is equidistant from its three cities L, M, and N. Which construction correctly finds the best location of the hospital?

Construction X is triangle LMN with an inscribed circle. Point A is on segment LM, point D is on segment MN, and point B is on segment LN, segment AN bisects angle N, segment MN bisects angle M, the angle formed by MDL is a right angle, and point C is the intersection of segments MB, LD, and AN.
Construction X
Construction Y is triangle LMN within a circumscribed circle. Point A is on segment LM, point B is on segment MN, point D is on segment MB, point C is on segment LN, angle MAB is a right angle, angle NCD is a right angle, and point E is the intersection of segments AB and EC.
Construction Y

Construction X because point C is the incenter of ΔLMN
Construction X because point C is the circumcenter of the ΔLMN
Construction Y because point E is the incenter of ΔLMN
Construction Y because point E is the circumcenter of ΔLMN