matematicas visuales visual math
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)

Wallace-Simson lines
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:

Simson Line, Simson-Wallace Line: a demonstration | matematicasVisuales

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.

Simson Line, Simson-Wallace Line: a demonstration | matematicasVisuales
Simson Line, Simson-Wallace Line: a demonstration | matematicasVisuales

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

Simson Line, Simson-Wallace Line: a demonstration | matematicasVisuales

Two right triangles are similar, then:

Simson Line, Simson-Wallace Line: a demonstration | matematicasVisuales

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

Simson Line, Simson-Wallace Line: a demonstration | matematicasVisuales

Simson Line, Simson-Wallace Line: a demonstration | matematicasVisuales

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

Simson Line, Simson-Wallace Line: a demonstration | matematicasVisuales

Simson Line, Simson-Wallace Line: a demonstration | matematicasVisuales

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.

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Central and inscribed angles in a circle | Mostration | Case II
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