P-circle

The common incircle of the medial triangle Δ M_A M_B M_C and the congruent triangle Δ Q_A Q_B Q_C, where Q_i are the midpoints of the line segment joining the Nagel point Na with the vertices of the original triangle Δ A B C. The Spieker circle has circle function l = - (-3 a^2 + 5b^2 + 5c^2 - 6b c + 2a b + 2a c)/(16b c), which does not correspond to any named center. The center of the Spieker circle is called the Spieker center Sp, and the circle has radius R_S = 1/2 r, where r is the inradius and s is the semiperimeter of the reference triangle.

incircle | medial triangle | midpoint | Nagel point | Spieker center