WallaceSimson

On the Wallace-Simson line

Consider the triangle ABC and a point D on its circumcircle. The projections E, F, G of D on the sides of the triangle are on a line L(D), the Wallace-Simson line of D. There are three cyclic quadrangles GDEA, DEFC and BFDG. Their centers lie on a circle (c) passing through D and J, the circumcenter of ABC. JD being a diameter, the circle is tangent and has half the radius of the circumcircle of ABC. The triangle LHI, with vertices the centers of these three circles is inscribed in (c) and is homothetic to ABC from D, with ratio 1/2. L, H, I are respectively the middles of segments AD, BD and CD. L(D) passes through the orthocenter M of LHI, being its Steiner line S(D) (look at SteinerLine.html ). The orthocenter N of ABC, M and D are on a line and M is the middle of ND. Points L, H and I move, as D glides on the circumference, on circles with diameters JA, JB and JC respectively. The line PQ of intersection of the euler circle (e) of ABC and (c) envelopes an ellipse (a conic in general) with foci at O (center of e) and J (circumcenter) and major circle the Euler circle (e).

The statement of the five points L, H, J, I and D lying on circle (c) is a consequence of Miquel's theorem (see the file Miquel_Point.html ). The other statements follow easily from this.

The statement on the ellipse is a particular case of a subject handled in the file(s) ThreeCirclesProblem.html and subsequent files referred there.

For the definition of the Simson-Wallace line look at the file Simson.html .

On the Wallace-Simson line

See Also