[justify][i]En un triángulo equilátero la suma de las distancias de un punto interior a los lados es constante.[br][br][/i]Para comprobar que este teorema se cumple, comenzamos dibujando un triángulo equilátero y un punto interior P.[br][i]A continuación, con ayuda de la herramienta [b]Recta perpendicular [/b]trazamos las rectas perpendiculares a cada lado que pasan por P.[br][img 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distancia a cada lado quedará fijada por el punto de intersección de cada una de estas rectas con su[br]respectivo lado.[br]Utilizando [b]Intersección[/b] encontramos cada uno de los puntos.[br][img width=284,height=213]data:image/png;base64,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[/img][br]Las tres distancias serán los segmentos PD, PE y PF que definimos como segmentos, ocultando previamente las tres rectas.[br][img 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continuación, tenemos que obtener la suma de estos tres segmentos; a los que previamente haremos que aparezca su nombre, pulsando el botón derecho sobre cada uno de ellos para que aparezca la opción [b]Mostrar etiqueta[/b].[br][img 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suma de los tres segmentos la obtendremos y expondremos en la vista gráfica con ayuda de la  herramienta [b]Texto[/b] [url=http://wiki.geogebra.org/es/Archivo:Tool_Insert_Text.gif][img width=33,height=33]data:image/png;base64,R0lGODlhIAAgAPABAAAAAP///yH/C01TT0ZGSUNFOS4wGAAAAAxjbVBQSkNtcDA3MTIAAAAHT223pQAsAAAAACAAIAAAAkqMj6nL7Q+jnLTai7PevPsFJMBIimUThoe6ti4YsLFoyDWTIrZqoy95mkVYRJrwATztbpDlkdlz0qSyKCy5C3623K73Cw6Lx2RDAQA7[/img][/url].[br]

Este valor constante coincide con la altura del triángulo ya que cuando P es un vértice del triángulo, la suma de las distancias a los lados es la altura.[br]Podemos finalizar la comprobación trazando y midiendo la altura.[br][br]