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 tracĂ©es depuis les sommets d’un carrĂ© vers les cĂŽtĂ©s opposĂ©s sont concourantes en un point Ă  l’intĂ©rieur du carrĂ©, nous pouvons utiliser la rotation gĂ©omĂ©trique. ConsidĂ©rons un carrĂ© ABCD contenant un point P. Nous montrerons que les droites perpendiculaires tracĂ©es 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 Ă  BP. Appelons X l'intersection de cette ligne perpendiculaire et de BP. 2. Effectuons maintenant une rotation du carrĂ© de 90 degrĂ©s 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 mappĂ© au segment de droite. CP. 3. AprĂšs la rotation, la droite perpendiculaire de C Ă  CP coĂŻncide avec la droite perpendiculaire d’origine de B Ă  BP. Par consĂ©quent, le carrĂ© pivotĂ© ABCD a la mĂȘme propriĂ©té : les droites perpendiculaires allant de B, C et D Ă  CP, DP et AP, respectivement, se coupent Ă©galement au point X. 4. RĂ©pĂ©tez le processus pour les sommets restants du carrĂ©. Effectuez successivement des rotations de 90 degrĂ©s dans le sens des aiguilles d’une montre autour des points B, C et D. Chaque rotation mappe le carrĂ© sur lui-mĂȘme et prĂ©serve la propriĂ©tĂ© des lignes perpendiculaires concurrentes. Par consĂ©quent, les lignes perpendiculaires allant de A, B, C et D Ă  BP, CP, DP et AP, respectivement, se coupent toutes au point X, qui est le point d'intersection de toutes les lignes perpendiculaires pivotĂ©es. Ainsi, nous avons montrĂ© que les lignes perpendiculaires tracĂ©es depuis les sommets A, B, C et D du carrĂ© jusqu’aux cĂŽtĂ©s opposĂ©s BP, CP, DP et AP, respectivement, sont concourantes en un point Ă  l’intĂ©rieur du carrĂ©.

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