An interesting equivalence with nice consequencies.

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An interesting equivalence with nice consequencies.

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#1 h Jun 21, 2009, 11:35 PM • 2 Y

Y by Adventure10, Mango247

Let $\{M,A,I,B,N\}$ be five different points so that $\left\|\begin{array}{c} IA = IB = r \\ \\ AM\perp AI \\ \\ BN\perp BI\end{array}\right\|$ .

Prove that $\boxed {\ IM\perp AN\ \Longleftrightarrow\ AM^2 + BN^2 = MN^2\ \Longleftrightarrow\ IN\perp BM\ }$ .

Particular cases.

$1\blacktriangleright$ Let $ABCD$ be a trapezoid so that $AD\parallel BC$ and $AB\perp AD$ . Denote the

midpoint $I$ of $[AB]$ . Then $\boxed {\ CI\perp BD\Longleftrightarrow DI\perp CA\Longleftrightarrow BC^2 + AD^2 = CD^2\ }$ .

Proof. Choose $\{D,A,I,B,C\}$ in the above general problem.

$2\blacktriangleright$ The incircle $w = C(I,r)$ of $\triangle ABC$ touches the sides $[BC]$ , $[CA]$ , $[AB]$ at $D$ , $E$ , $F$

respectively. Denote $P\in BC\cap EF$ . Then $PI\perp AD$ and $PD^2 + AE^2 = PA^2$ (see here).

Proof. Choose $\{A,E,I,D,P\}$ in the above general problem.

$3\blacktriangleright$ The $A$ -exincircle $w = C(I_a,r_a)$ of $\triangle ABC$ touches $BC$ , $CA$ , $AB$ at $D_a$ , $E_a$ , $F_a$

respectively. Denote $P_a\in BC\cap E_aF_a$ . Then $P_aI_a\perp AD_a$ and $P_aD_a^2 + AE_a^2 = P_aA^2$ .

Proof. Choose $\{A,E_a,I_a,D_a,P_a\}$ in the above general problem.

This post has been edited 1 time. Last edited by Virgil Nicula, Jun 22, 2009, 5:40 PM

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#2 h Jun 22, 2009, 2:58 AM • 2 Y

Y by Adventure10, Mango247

Virgil Nicula wrote:

Let $\{I,A,B,M,N\}$ be five different points so that $\left\|\begin{array}{c} IA = IB = r \\ \\ AM\perp AI \\ \\ BN\perp BI\end{array}\right\|$ .

Prove that $\boxed {\ IM\perp AN\ \Longleftrightarrow\ AM^2 + BN^2 = MN^2\ \Longleftrightarrow\ IN\perp BM\ }$ .

Very interesting lemma !
Let me solve it

$MN^2 = AM^2 + BN^2\Leftrightarrow MN^2 = AM^2 + IN^2 - IB^2$
$\Leftrightarrow MN^2 - IN^2 = AM^2 - IA^2\Leftrightarrow NA\perp MI$
Similarly, $MB\perp IN$

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#3 h Jun 23, 2009, 10:17 PM • 2 Y

Y by Adventure10, Mango247

See here another application of this problem !

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