let be $ ABCDE $ the pentagon and $ X,Y,Z,T,W $ are the intersection of the pairs of lines $ (AE,BC),(AB,DC),(BC,DE),(DC,AE),(DE,AB) $. the angles of the triangles $ ABX,AEW $ are congruent. the angle $ AEW $ is the sum between the angles $ EXZ $ and $ EZX $ so the angle $ AEW $ is congruent with the angle $ ABX $. so the angles $ B,E $ of the pentagon are congruent and $ AB=AE $. in similar ways we'll obtain all the angles are congruent and also all the sides so the pentagon will be regular.