Bottema's Point Sibling

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Problem

Bottema's Point Sibling, problem

Solution

  1. We prove that triangles $BCB_c\,$ and $A_cCA\,$ are equal. (Thus $AA_c=BB_c,$ as a consequence.) Indeed, $CB_c=CA,\,$ $CA_c=CB,\,$ and $\angle BCB_c=90^{\circ}+\angle ACB=\angle ACA_c.\,$ One is obtained from the other with a rotation through $90^{\circ}\,$ around point $C,\,$ making corresponding sides perpendicular. In particular, $AA_c\perp BB_c.$ Let $N\,$ be the intersection of the two.

  2. Let $M\,$ be Bottema's point. Among the many properties it has, $M\,$ lies on the circle with diameter $AB.\,$ Since $\angle ANB=90^{\circ},\,$ $N\,$ lies on the same circle. $M\,$ is known to divide the arc $\overset{\frown}{AB}\,$ to which it belongs in half. This means that the arc $overset{\frown}{BM}\,$ is one quarter of the circle, making inscribed $\angle BNM=45{\circ}.$

    Bottema's Point Sibling, 2

    $N\,$ also belongs to the circle $(CBA_bA_c)\,$ because $\angle BNA_c=90^{\circ}\,$ and $BA_c\,$ is a diameter of that circle. Inscribed $\angle BNA_b=45^{\circ}\,$ for it too is subtended by a quarter of a circle. Thus $M\,$ lies on $NA_b.$

    Bottema's Point Sibling, 3

    Similarly, $N\in (ACB_cB_a)\,$ and, for that reason, $\angle ANB_a=45^{\circ},\,$ implying that A_bNB_a is a straight line. Thus the three lines $AA_c,BB_c,A_bB_a\,$ are indeed concurrent and

  3. $A_bB_a\,$ is the bisector of angles $ANB_c\,$ and $BNA_c,$

  4. Inscribed $angle CNA_b=90^{\circ},\,$ as inscribed and subtended by the diameter $CA_b.\,$ If $M'\,$ is the second intersection of $CN\,$ with $(ANB),\,$ $\angle MNM'=90^{\circ},\,$ making $MM'\,$ a diameter of $(ANB).$

    Bottema's Point Sibling, 4

Acknowledgment

The problem has been tweeted by Antonio Gutierrez and introduced with a GeoGebra applet by Tim Brzezinski.

The configuration is a part of Vecten's construction involving squares on each side of a triangle that is known as Bride's chair. Here we relate it to the famous Bottema's problem.

 

Bottema's Theorem

  1. Bottema's Theorem
  2. An Elementary Proof of Bottema's Theorem
  3. Bottema's Theorem - Proof Without Words
  4. On Bottema's Shoulders
  5. On Bottema's Shoulders II
  6. On Bottema's Shoulders with a Ladder
  7. Friendly Kiepert's Perspectors
  8. Bottema Shatters Japan's Seclusion
  9. Rotations in Disguise
  10. Four Hinged Squares
  11. Four Hinged Squares, Solution with Complex Numbers
  12. Pythagoras' from Bottema's
  13. A Degenerate Case of Bottema's Configuration
  14. Properties of Flank Triangles
  15. Analytic Proof of Bottema's Theorem
  16. Yet Another Generalization of Bottema's Theorem
  17. Bottema with a Product of Rotations
  18. Bottema with Similar Triangles
  19. Bottema in Three Rotations
  20. Bottema's Point Sibling

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