Lines of Simson-Wallace: Demonstración

We can start with a triangle and its circunscribed circle. Given a point P on the circumcircle of a triangle, the feet of the perpendiculars from P to the three sides all lie on a straight line (Simson line or Simson-Wallace line)

Each point in the circle circunscribed to a triangle give us a line (Wallace-Simson line)

We are going to see this property using this notation:

We have taken P to lie on the arc AC that does not contain B. Other cases can be derived by re-naming A, B, C.

If we can prove that these two angles are equal then points A', B', C' will be collinear.

We can use a consequence of a circle property (Euclides, III.21 or III.22) that saids that the opposite angles of every convex cuadrangle inscribed in a circle are together equal to two right angles.

Central angle in a circle is twice the angle inscribed in the circle.

Two right triangles are similar, then:

Points A, B', P, C' lies on a circle:

And points B',A',C,P lies on a circle:

Then points A', B', C' are collinear. This is called Simson Line or Simson-Wallace Line of P.

REFERENCES

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: John Wiley and sons, 1969.
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer.