Lines of SimsonWallace: 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 SimsonWallace 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 renaming 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.
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 SimsonWallace 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.
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.
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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.
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