## 1. A parallel to the Simson-Wallace line

Consider triangle t=ABC and a point D on its circumcircle. Project D on the sides of (t) and define the Simson-Wallace line W(D) (see Simson.html ) through these projections: E, F and G. Extend DE to define its intersection point H with the circumcircle. Then CH is parallel to the Simson line W(D).

The proof follows at once from the fact that quadrangles DFEB and DCHB are cyclic. Hence angle(CHD) = angle(CBD) = angle(HEF). Showing that EF and CH are parallel.
This has an interesting consequence studied in SimsonProperty2.html .
Remark Another application is the determination of a point D, such that its Simson line S(D) has a given direction:
Draw from C a line CH in the given direction and determine point H. Then from H draw the orthogonal to the oposite side (AB) intersecting the circle at a second point D. This is the desired point.

## 2. A related parallelogram

From a point P on the circumcircle of triangle ABC draw PDF orthogonal to side BC, F being the second intersection point with the circumcircle. Extend the altitude HA by AG = DF, H being the orthocenter. Then GP = HD or equivalently HDPG is a parallelogram.

The proof follows from the equality of triangles DEF and GPA which is trivial and the fact that E is symmetric to H with respect to BC.
Remark Since by the previous section DG is the Simson-Wallace line of P, the middle M of PH is on this line and simultaneously on the Euler circle of ABC, which is homothetic to the circumcircle with respect to H and in ratio 1/2. This gives another argument to prove the property of the Steiner line parallel to DG (see SteinerLine.html ).
This argument is used also to prove a property of orthopoles (see Orthopole.html ).

Deltoid.html
Orthocenter.html
Orthopole.html
Projection_Triangle.html
Simson.html
SimsonDiametral.html
SimsonGeneral.html
SimsonGeneral2.html
SimsonProperty2.html
Simson_3Lines.html
SimsonVariantLocus.html
SteinerLine.html
ThreeDiameters.html
ThreeSimson.html
TrianglesCircumscribingParabolas.html
WallaceSimson.html

### References

Lalesco, T. La Geometrie du Triangle. Paris, Jacques Gabay, 1987, p. 8