An Area property of Triangles

Moderator: elim

An Area property of Triangles

Postby elim » Wed Aug 31, 2011 12:42 pm

Let point $P$ be inside of $\Delta ABC$, and 3 lines through $\{A,P\ \},\{B,P\ \}$, and $\{C,P\ \} $ intersect
$\Delta ABC $ at $A_1, B_1, C_1$ respectively. Let $A_2, B_2, C_2$ be middle points of $\overline{AA}_1,\ \overline{BB}_1$ and $\overline{CC}_1$.

Show that $\ |\Delta A_1 B_1 C_1|= 4|\Delta A_2 B_2 C_2|$

001.gif
001.gif (8.87 KiB) Viewed 418 times
elim
 
Posts: 64
Joined: Mon Feb 08, 2010 8:03 pm

Re: An Area property of Triangles

Postby elim » Wed Aug 31, 2011 1:07 pm

Let $A_3, B_3, C_3$ be the symmetric points of P on $\overline{AA}_1,\overline{BB}_1, \ \overline{CC}_1\ $ with respect to their middle points.

Show that $|\Delta A_1 B_1 C_1| = |\Delta A_3 B_3 C_3|$

002.gif
002.gif (10.84 KiB) Viewed 417 times
elim
 
Posts: 64
Joined: Mon Feb 08, 2010 8:03 pm

Re: An Area property of Triangles

Postby elim » Wed Aug 31, 2011 1:52 pm

Note that $[ACC_3A_3]=[APC_1]+[PA_1C_3]$. Thus $[ABC]-[A_3B_3C_3]=[C_1AA_3B_1CC_3A_1BB_3]$
so $[A_3B_3C_3]=[AA_3B_1]+[BB_3C_1]+[CC_3A_1]=[PA_1B_1]+[PB_1C_1]+[PC_1A_1]=[A_1B_1C_1]$
elim
 
Posts: 64
Joined: Mon Feb 08, 2010 8:03 pm

Re: An Area property of Triangles

Postby Bataouche » Mon Sep 10, 2012 1:28 pm

It's really a cool and helpful piece of information. I'm satisfied that you shared this helpful information with us. Please stay us informed like this. Thanks for sharing it on elinkage.net wish you luck!
Bataouche
 
Posts: 1
Joined: Mon Sep 10, 2012 1:22 pm


Return to Geometry

Who is online

Users browsing this forum: No registered users and 1 guest

cron