### Parallel Chords II: What is this about?

A Mathematical Droodle

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Copyright © 1996-2018 Alexander Bogomolny

The applet purports to suggest the following statement [F. G.-M., Theorem 124, 639°]:

A family of coaxal circles passes through the vertices B and C of ΔABC. P and Q are the second points of intersection (other than B and C) of the circles with sides AB and AC. Then all lines PQ are parallel. |

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The statement is actually trivial: since the quadrilateral BCQP is cyclic with fixed angles at B and C, the angles at P and Q are also fixed. This is unconditionally true for all pairs P, Q on the same side from A. Some caution must be exercised when combining the two cases.

We may note that the chords in question are parallel to the side of the tangential triangle through vertex A.

Also, the statement admits a converse of sorts: assume a quadrilateral BCQP is cyclic as above. Assume that M on AB and N on AC are such that MN||PQ. Then the quadrilateral BCNM is cyclic.

### References

- F. G.-M.,
*Exercices de Géométrie*, Éditions Jacques Gabay, sixiéme édition, 1991

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