The Steiner deltoid is the envelope of the Simson lines of a triangle. Its circumcircle is the Steiner circle, and its incircle is the nine-point circle. The triangle formed by vertices of the deltoid is homothetic to the first Morley triangle, with the center dividing the line X_5 X_356 in the ratio of their side lengths, k = 3/16 sqrt(3)csc(1/3 A) csc(1/3 B) csc(1/3 C). The triangle formed by the deltoid's meets with the nine-point circle is also is homothetic to the first Morley triangle, with the center dividing the line X_5 X_356 in the ratio of their side lengths k = - 1/16 sqrt(3)csc(1/3 A) csc(1/3 B) csc(1/3 C).

deltoid | first Morley triangle | Simson line | Steiner circle