SimsonDiametral

Simson lines of diametral points

Consider triangle t=ABC and a point D on its circumcircle (c) and its antipodal point E on (c). Project D, E on the sides of (t) and define the Simson lines S(D), S(E) (see Simson.html ). Then the angle of the two Simson lines S(D) and S(E) is a right one and their intersection point is on the Euler circle of t.

That the angle of S(D), S(E) is a right one follows from the discussion in SimsonProperty2.html .
The proof of the other statement follows from the fact that the orthocenter H of t is the similarity center of the Euler circle and the circumcircle of t. To complete the argument draw parallels to S(D), S(E) intersecting at point G on (c). From SteinerLine.html follows that the intersection point F of S(D) and S(E) and the intersections J, I of these lines with HD, HE make a triangle s = IJF, similar to DEF etc. Notice, by the way that diameter IJ of the Euler circle is parallel to the diameter DE of the circumcircle c.

Remark Projecting the points {E,O,D} of the diameter on one side, AB say, we see that the projections of {D,E} are symmetric with respect to the middle of the corresponding side. This could be used to give another proof of the property discussed above.

Simson lines of diametral points

See Also