Question

22. Let [AB] be a chord in a circle C, and k a circle which is internally tangent to the circle C at a point P and to the chord [AB] at a point Q. Show that the line P Q passes through the midpoint of the arc AB opposite to the arc APB.

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Answer to a math question 22. Let [AB] be a chord in a circle C, and k a circle which is internally tangent to the circle C at a point P and to the chord [AB] at a point Q. Show that the line P Q passes through the midpoint of the arc AB opposite to the arc APB.

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Dexter
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Pour prouver que la droite PQ passe par le milieu de l’arc AB opposé à l’arc APB, on peut suivre ces étapes : 1. Soit O le centre du cercle C, et M le milieu de l'arc AB opposé à l'arc APB. 2. Puisque le cercle k est intérieurement tangent au cercle C au point P, nous savons que la droite OP est perpendiculaire à PQ. En effet, le rayon du cercle C au point P est perpendiculaire à toute ligne tangente passant par P. 3. Soit N le point d'intersection de la droite PQ et du cercle C. Il faut montrer que N est identique à M, milieu de l'arc AB opposé à l'arc APB. 4. Puisque OP est perpendiculaire à PQ, le triangle OPQ est un triangle rectangle. 5. Considérant le triangle rectangle OPQ, nous savons que l’hypoténuse OQ est un diamètre du cercle C. L’angle OQP est donc un angle droit. 6. Puisque l’angle OQP est un angle droit, et que l’angle OMP est également un angle droit (puisque M est le milieu de l’arc AB), on peut conclure que le quadrilatère OMNQ est un quadrilatère cyclique. 7. Par les propriétés d'un quadrilatère cyclique, les angles opposés de OMNQ sont supplémentaires. L’angle OMN est donc complémentaire à l’angle OQN. 8. Puisque l'angle OMN est supplémentaire à l'angle OQN, et que l'angle OQN est un angle droit, il s'ensuit que l'angle OMN est aussi un angle droit. 9. L'angle OMN étant un angle droit signifie que MN est perpendiculaire à la corde AB. 10. Puisque MN est perpendiculaire à la corde AB et que M est le milieu de l’arc AB opposé à l’arc APB, on peut conclure que la droite PQ passe par le milieu M de l’arc AB opposé à l’arc APB. Nous avons donc montré que la droite PQ passe par le milieu de l’arc AB opposé à l’arc APB.

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