54. Cinci grade de libertate.

by Virgil Nicula, Jul 9, 2010, 10:50 PM

PP. Let $\ d_0$ be a fixed straight line and $A$ , $B$ , $C$ , $D$ , $E$ which belong to line $\ d_0$ . Through $A$ passes a mobile $\delta$ on which

we choose two mobile $M$ , $N$ . Let $\ U\in BM\cap DN$ and $V\in CM\cap EN$ . Prove that $UV$ passes through a fixed point.

Proof 1. Let $S\in UV\cap d_0$ . Apply the Menelaus' theorem to $\overline{MVC}/\triangle ANE$ , $\overline{MUB}/\triangle ADN$ , $\overline{SUV}/\triangle DNE\ :\ \frac{VE}{VN}\cdot$ $\frac{MN}{MA}\cdot\frac{CA}{CE}=1\ ;$

$\frac{UN}{UD}\cdot\frac{BD}{BA}\cdot\frac{MA}{MN}=1$ ; $\frac{SE}{SD}\cdot\frac{UD}{UN}\cdot\frac{VN}{VE}=1$ . From the product of these relations obtain the relation $\frac{SE}{SD}\cdot\frac{CA}{CE}\cdot\frac{BD}{BA}=1$ , i.e. $\frac{SE}{SD}=\frac{BA}{BD}\cdot\frac{CE}{CA}$ .

Thus the point $S$ is fixed. Prove analogously that $\frac{SB}{SC}=\frac{DB}{DA}\cdot\frac{EA}{EC}$ . From the product of the last two relations obtain the relation $\frac{SD}{SE}\cdot\frac{AB}{AC}=\frac{SB}{SC}\cdot\frac{AD}{AE}$ .

Proof 2. Let $\left\{\begin{array}{ccccc} A(0,0)\ ;\ B(b,0)\ ;\ C(c,0)\\\ D(d,0)\ ;\ E(a,0)\ ;\ S(x,0)\end{array}\right\|$ , where $a$ , $b$ , $c$ , $d$ are different two by two. The equation of $\delta$ is $y=px$ . $U\in BM\cap DN$ $\implies$ $x_u=\frac{mn(b-d)-bd(m-n)}{nb-md}$ and

$y_u=\frac{pmn(b-d)}{nb-md}$ . $V\in CM\cap EN$ $\implies$ $x_v=\frac{nm(a-c)-ac(n-m)}{ma-nc}$ and $y_v=\frac{pnm(a-c)}{ma-nc}$ . Thus, $\left|\begin{array}{ccc}x_u&y_u&1\\\\x_v&y_v&1\\\\x&0&1\end{array}\right|=0$ $\Longleftrightarrow$ $\left|\begin{array}{ccc}bd(n-m)&b-d&d\\\\ac(m-n)&a-c&a(m-n)\\\\x&0&1\end{array}\right|=0$

$\Longleftrightarrow$ $x=\frac{bd(a-c)+ac(b-d)}{ab-cd}$ (constant). The conditions $ab\ \neq\ cd,\ i.e.\ AB\cdot AE\ \neq\ AC\cdot AD$ and $\frac{m}{n}\ \not\in \ \left\{\ \frac{b}{d},\ \frac{c}{a}\ \right\}$ what mean synthetically ?
This post has been edited 21 times. Last edited by Virgil Nicula, Feb 9, 2016, 7:23 AM

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