A Circle With Two Centers
How Is It Possible?

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A Circle With Two Centers

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At points B and C of an angle BAC erect perpendiculars and let them intersect in D. Since AB and AC are not parallel, the perpendiculars are not parallel either. Thus, unless AB and AC form a 0° or a 180° angle, point D is well defined.

Draw a circle circumscribing ΔBCD. Besides B, the circle intersects AB in another point, say, E. It intersects AC in F different from C. Now, since ∠DBE is right, DE is a diameter of the circle. The midpoint of DE is the center of the circle. A similar argument applies to DF. As a result, we arrive at the conclusion that the circle at hand has two centers. How come?

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Copyright © 1996-2018 Alexander Bogomolny

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What if applet does not run?

The problem is with the drawing. Since both angles ABD and ACD are right, they both subtend a diameter of the circle, which is AD. Thus in all cases the points E and F coincide with A. (The applet cheats by drawing BD and CD not quite perpendicular to AB and AC.)

References

  1. V. M. Bradis et al, Lapses in Mathematical Reasoning, Dover, 1999, pp. 137-138

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Geometric Fallacies

  • Rouse Ball's Fallacy
  • All Triangles Are Isosceles
  • Two Perpendiculars From a Point to a Line
  • Is Every Trapezoid Parallelogram?
  • Every Parallelogram Is a Rectangle

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    Copyright © 1996-2018 Alexander Bogomolny

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