OWN. O interesanta echivalenta cu frumoase consecinte.

[Virgil Nicula]

Consideram cinci puncte diferite intre ele \( \{M,A,I,B,N\} \) astfel incat \( \left\|\begin{array}{c}
IA=IB\\\\\\\\\
AM\perp AI\\\\\\\\
BN\perp BI\end{array}\right\| \) .

Sa se arate ca \( \underline{\overline{\left\|\ \ IM\perp AN\ \Longleftrightarrow\ AM^2+BN^2=MN^2\ \Longleftrightarrow\ IN\perp BM\ \right\|}} \) .

[Mateescu Constantin]

Lema:

Daca \( A \) , \( B \) , \( C \) , \( D \) sunt patru puncte din plan atunci \( \fbox{\ AC\perp BD\ \Longleftrightarrow\ AB^2+CD^2=AD^2+BC^2\ } \) .

Aplicand lema \( \Longrightarrow\ \fbox{\ IM\perp AN\ }\ \Longleftrightarrow\ IN^2+AM^2=IA^2+MN^2\ \Longleftrightarrow\ IN^2-IA^2+AM^2=MN^2\ \)

\( \Longleftrightarrow\ IN^2-IB^2+AM^2=MN^2\ \Longleftrightarrow\ \fbox{\ AM^2+BN^2=MN^2\ }\ \Longleftrightarrow\ IM^2-IA^2+BN^2=MN^2 \)

\( \Longleftrightarrow\ MI^2+BN^2=IB^2+MN^2\ \Longleftrightarrow\ \fbox{\ IN\perp BM\ } \) .