Concurrent Chords in a Circle, Equally Inclined

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Concurrent Chords in a Circle, Equally Inclined, problem


Vladimir Dubrovsky
30 March, 2017

We'll make use of the Lagrange formula for the moment of inertia, also known as Steiner's and Huygens-Steiner theorem:

For unit masses placed at points $A_1,A_2,\ldots,A_n,\,$ the moment of inertia with respect to a point $X\,$ is defined as $\displaystyle I_X=\sum_{k=1}^nXA_k^2,\,$ and the Lagrange formula takes the form $\displaystyle I_X=nXG^2+I_G,\,$ where $G\,$ is the centroid (center of mass) of these material points.

In particular, $I_X,\,$ as a function of $X,\,$ depends only on the distance $XG.$

The centroid of the six endpoints of the given chords coincides with the centroid of the midpoints of the chords, which are the projections of $O\,$ - the center of the given circle - on the chords, and hence form a triangle inscribed in the circle $\mathbb{c}\,$ with diameter $BO.\,$ By the Inscribed Angle Theorem, the triangle is equilateral.

Concurrent Chords in a Circle, Equally Inclined

Therefore, its centroid $G\,$ is the center of circle $\mathbb{c},\,$ or the midpoint of $BO.\,$ It follows that $BG=GO,\,$ so, by the Lagrange formula,

$\displaystyle BN^2+BM^2+BP^2+BQ^2+BR^2+BS^2=I_B=I_G=6R^2$

and we are done.

This proof readily generalizes to $n\ge 3\,$ chords drawn through the same point at equal angles to each other: the sum of the squares of their pieces is equal to $nR^2.$


The problem has been discussed elsewhere as one of the properties (viz., #3) of the configuration of six concurrent chords. Vladimir Dubrovsky has commented with his solution, reproduced above.


Barycenter and Barycentric Coordinates

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  2. Barycentric Coordinates
  3. Barycentric Coordinates: a Tool
  4. Barycentric Coordinates and Geometric Probability
  5. Ceva's Theorem
  6. Determinants, Area, and Barycentric Coordinates
  7. Maxwell Theorem via the Center of Gravity
  8. Bimedians in a Quadrilateral
  9. Simultaneous Generalization of the Theorems of Ceva and Menelaus
  10. Three glasses puzzle
  11. Van Obel Theorem and Barycentric Coordinates
  12. 1961 IMO, Problem 4. An exercise in barycentric coordinates
  13. Centroids in Polygon
  14. Center of Gravity and Motion of Material Points
  15. Isotomic Reciprocity
  16. An Affine Property of Barycenter
  17. Problem in Direct Similarity
  18. Circles in Barycentric Coordinates
  19. Barycenter of Cevian Triangle
  20. Concurrent Chords in a Circle, Equally Inclined

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