We can start with a triangle and its circunscribed circle. If P is any point belonging to this circuncircle: Then we can consider the feet of the perpendiculars from P to the three sides of the triangle (these are ortogonal projections of a point onto the sides or their prolongations): These three feet of the perpendiculars lie on a line: This line is called the Simson Line of P respect to the triangle or WallaceSimson Line. The first to mention this line was Wallace (17681843), in 1799, thirty years after Simson's death (16871768).
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.
de Guzmán, Miguel 'The envelope of the WallaceSimson lines of a triangle. A simple proof of the Steiner theorem on the deltoid'.
RACSAM, vol. 95, 2001.
Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, 1965.
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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.
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.
Interactive 'Mostation' of the property of central and inscribed angles in a circle. The general case is proved.
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.
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)
