SimsonGeneral2

Generalized Simson Line

From a point J of the circumcircle of triangle ABC draw lines at an angle (phi) to the sides of the triangle. The intersection-points of these lines with the respective sides of the triangle are on a line depending on the position of J. For (phi) equal to (pi)/2 this is the well known theorem of Simson.

The generalized Simson lines, for various positions of the point J on the circumcircle of ABC, envelope a deltoid [d] similar to the envelope of the Simson lines of ABC. The tangent circle of [d] is equal to the Miquel circle of points J, L, M, etc. (look at SimsonGeneral.html ). This tangent circle results by applying a spiral similarity F on the Euler circle of ABC (is tangent to the corresponding deltoid). F has center at the circumcenter L of ABC, angle of rotation (psi) = (pi)/2 - (phi) and modulus 1/cos(psi) = LR/LK.

Generalized Simson Line

See Also