A round coin may be used to construct a circle passing through one or two given points on the plane. Given a line on the plane, show how to use this coin to construct two points such that they dene a line perpendicular to the given line. Note that the coin may not be used to construct a circle tangent to the given line.
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Tags: geometry unsolved, geometry
SomePig
13.02.2011 13:55
[geogebra]ce27d0952fd9d698d6160810886674ebf3029cf7[/geogebra]
baenanabread
07.08.2020 23:37
There are two distinct congruent circles passing through any two points X and Y, and one is a reflection of the other through the line XY. This can be showed by naming their centers O and O', either on opposite sides of the plane with respect to XY. Then OXO'Y is clearly a rhombus and their centers are a reflection of each other and thus the circles themselves are. Pick adequate points C', B, C, B' on the line L and in that order. Draw two circles P and Q, going through C, C' and B, B' respectively. Let A be one of their intersections. Now construct the congruent circles P' and Q' going through the same points as their homologous circles. Name A' the intersection point of P' and Q' located on the opposite side of the plane to A. As we showed, P' and Q' are the reflections of P and Q with respect to L and thus, corresponding points A and A' are a reflection of each other and the line AA' is perpendicular to L.