Question

P 13. Let P a point inside of a square ABCD. Show that the perpendicular lines drawn from A, B, C, respectively D, to BP, CP, DP, respectively AP are concurrent. Use geometric rotation.

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Answer to a math question P 13. Let P a point inside of a square ABCD. Show that the perpendicular lines drawn from A, B, C, respectively D, to BP, CP, DP, respectively AP are concurrent. Use geometric rotation.

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Gene
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Pour montrer que les lignes perpendiculaires tracées depuis les sommets d’un carré vers les côtés opposés sont concourantes en un point à l’intérieur du carré, nous pouvons utiliser la rotation géométrique. Considérons un carré ABCD contenant un point P. Nous montrerons que les droites perpendiculaires tracées respectivement de A, B, C et D vers BP, CP, DP et AP se coupent en un seul point. 1. Dessinez le segment de droite AP et construisez une ligne perpendiculaire de A à BP. Appelons X l'intersection de cette ligne perpendiculaire et de BP. 2. Effectuons maintenant une rotation du carré de 90 degrés dans le sens des aiguilles d'une montre autour du point A. Cette rotation mappe le point B au point C, le point C au point D et le point D au point P. Le segment de droite BP est mappé au segment de droite. CP. 3. Après la rotation, la droite perpendiculaire de C à CP coïncide avec la droite perpendiculaire d’origine de B à BP. Par conséquent, le carré pivoté ABCD a la même propriété : les droites perpendiculaires allant de B, C et D à CP, DP et AP, respectivement, se coupent également au point X. 4. Répétez le processus pour les sommets restants du carré. Effectuez successivement des rotations de 90 degrés dans le sens des aiguilles d’une montre autour des points B, C et D. Chaque rotation mappe le carré sur lui-même et préserve la propriété des lignes perpendiculaires concurrentes. Par conséquent, les lignes perpendiculaires allant de A, B, C et D à BP, CP, DP et AP, respectivement, se coupent toutes au point X, qui est le point d'intersection de toutes les lignes perpendiculaires pivotées. Ainsi, nous avons montré que les lignes perpendiculaires tracées depuis les sommets A, B, C et D du carré jusqu’aux côtés opposés BP, CP, DP et AP, respectivement, sont concourantes en un point à l’intérieur du carré.

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