## An Area property of Triangles

Moderator: elim

### An Area property of Triangles

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

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elim

Posts: 64
Joined: Mon Feb 08, 2010 8:03 pm

### Re: An Area property of Triangles

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 (10.84 KiB) Viewed 4593 times
elim

Posts: 64
Joined: Mon Feb 08, 2010 8:03 pm

### Re: An Area property of Triangles

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

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Bataouche

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Joined: Mon Sep 10, 2012 1:22 pm