matematicas visuales visual math

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.

MORE LINKS

Steiner deltoid is a hypocycloid
Steiner deltoid is a hypocycloid related with the nine point circle of a triangle.
The deltoid and the Morley triangle
Steiner Deltoid and the Morley triangle are related.
Central and inscribed angles in a circle
Central angle in a circle is twice the angle inscribed in the circle.
Central and inscribed angles in a circle | Mostration | Case I
Interactive 'Mostation' of the property of central and inscribed angles in a circle. Case I: When the arc is half a circle the inscribed angle is a right angle.
Central and inscribed angles in a circle | Mostration | Case II
Interactive 'Mostation' of the property of central and inscribed angles in a circle. Case II: When one chord that forms the inscribed angle is a diameter.
Central and inscribed angles in a circle | Mostration | General Case
Interactive 'Mostation' of the property of central and inscribed angles in a circle. The general case is proved.
Drawing fifteen degrees angles
Using a ruler and a compass we can draw fifteen degrees angles. These are basic examples of the central and inscribed in a circle angles property.
Morley Theorem
The three points of intersection of the adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle (Morley's triangle)
John Conway's proof of Morley's Theorem
Interactive animation about John Conway's beautiful proof of Morley's Theorem