(A.Myakishev, 9--10) Given a triangle $ ABC$. Point $ A_1$ is chosen on the ray $ BA$ so that segments $ BA_1$ and $ BC$ are equal. Point $ A_2$ is chosen on the ray $ CA$ so that segments $ CA_2$ and $ BC$ are equal. Points $ B_1$, $ B_2$ and $ C_1$, $ C_2$ are chosen similarly. Prove that lines $ A_1A_2$, $ B_1B_2$, $ C_1C_2$ are parallel.
Problem
Source: Sharygin contest 2008. The correspondence round. Problem 12
Tags: geometry proposed, geometry
moldovan
04.09.2008 20:14
Suppose W.L.O.G. that $ b>a>c$, where $ a,b,c$ are the usual notations. Let $ P=A_1A_2 \cap BC, B_1 \in AC$ and $ B_2 \in BC$. From Menelaus' theorem for the triangle $ ABC$ and the line $ A_1-A_2-P$ we deduce that: $ \frac {BP} {PC} \cdot \frac {a} {b-a} \cdot \frac {a-c} {a} =1 \Rightarrow \frac {BP} {PC}=\frac {b-a} {a-c} \Rightarrow PC=\frac {a(a-c)} {b-c}$. But also, it is easy to see that $ CB_1=b-c, B_2C=a-c$ and $ A_2C=a$. Finally, $ \frac {B_2C} {PC}=\frac {B_1C} {A_2C}$ and from the reciproc of Thales' theorem $ \Rightarrow A_1A_2 \parallel B_1B_2$.
armpist
05.09.2008 07:49
What makes all 3 of them parallel is the property of being simultaneously perpendicular to the IO line of triangle ABC. M.T.
stergiu
05.09.2008 10:04
armpist wrote: What makes all 3 of them parallel is the property of being simultaneously perpendicular to the IO line of triangle ABC. M.T. It seems nice , but why does this perpendicularity hold ? Babis
April
06.09.2008 11:37
stergiu wrote: armpist wrote: What makes all 3 of them parallel is the property of being simultaneously perpendicular to the IO line of triangle ABC. M.T. It seems nice , but why does this perpendicularity hold ? Babis http://www.mathlinks.ro/viewtopic.php?t=147540