Question

159. Intr-un triunghi echilateral, suma distanjelor unui punct luat în interiorul acestui triunghi la laturile lui este o mărime constantă (egală cu înălţimea triunghiului).​

Answer 1

$\it Fie\ M\in Int (\Delta ABC)\\ \\ MP=d(M,\ BC) \Rightarrow MP\perp BC \Rightarrow MP= \hat\imath n\breve a l\c{\it t}ime\ \hat\imath n\ \Delta MBC\\ \\ MQ=d(M,\ AC) \Rightarrow MQ\perp AC \Rightarrow MQ= \hat\imath n\breve a l\c{\it t}ime\ \hat\imath n\ \Delta MAC\\ \\ MR=d(M,\ AB) \Rightarrow MR\perp AB \Rightarrow MR= \hat\imath n\breve a l\c{\it t}ime\ \hat\imath n\ \Delta MAB$

$\it Fie \ \ AB=BC=AC=\ell;\ \ h=\hat\imath n\breve al\c{\it t}imea\ triunghiului\ ABC.\\ \\ \mathcal{A}_{ABC}=\dfrac{\ell\cdot h}{2}=\dfrac{\ell}{2}\cdot h\ \ \ \ \ (1)\\ \\ \\ \mathcal{A}_{ABC}= \mathcal{A}_{MBC}+ \mathcal{A}_{MAC}+ \mathcal{A}_{MAB}=\dfrac{\ell\cdot MP}{2}+\dfrac{\ell\cdot MQ}{2}+\dfrac{\ell\cdot MR}{2}=\\ \\ \\ =\dfrac{\ell}{2}\cdot(MP+MQ+MR)\ \ \ \ (2)\\ \\ \\ (1),\ (2) \Rightarrow MP+MQ+MR=h\ (constant)$