6. ABCD is a quadrilateral, and AC and BD are its diagonals. Is \( \mathrm{AB}+\mathrm{BC}+\mathrm{CD}+\mathrm{DA}>\mathrm{AC}+\mathrm{BD} \) ? 7. ABCD is a quadrilateral and \( O \) is any point in its interior in the given figure Show that \( A B+B C+C D+D A<2(O A+O B+O C+O D) \).

the area of \( \triangle A B C=12 \mathrm{~cm}^{2} \), then \( \left(b^{2}+c^{2}-a^{2}\right) \tan A=\cdots \) 12 (b) 24

\( A B C \) est un triangle et \( I \) milieu de \( [B C] \). 1) Soit \( \operatorname{Ebar}\{(A, 1) ;(B, 2) \),\( \} et F b a r\{(B, 2) ;,(C, 1)\} \). a) Montrer que: \( \overrightarrow{A E}=\frac{2}{3} \overrightarrow{A B} \). b) Montrer que: \( \overrightarrow{B F}=\frac{1}{3} \overrightarrow{B C} \) c) Construire les points \( I, E \) et \( F \). 2) Soit \( G b a r\{(A, 1) ;(B, 2) ;,(C, 1)\} \). a) Montrer que \( \operatorname{Gbar}\{(A, 1) ;(F, 3) \),\( \} . \) b) Montrer que \( G \in(E C) \). c) Montrer que \( G \in(I B) \) et construire \( G \). 3) Determiner l'ensemble des points \( M \) du plan qui verifient chacune des équations suivantes: a) \( \|\overrightarrow{M A}+2 \overrightarrow{M B}+\overrightarrow{M C}\|=8 \) b) \( \|\overrightarrow{M A}+2 \overrightarrow{M B}\|=\|\overrightarrow{M C}+2 \overrightarrow{M B}\| \)

e) La somma e la differenza delle diagonali di un rombo misurano 126 cm e 18 cm . Calcola la misura del raggio della circonferenza inscritta nel rombo.

6. \( A D, B E \) and \( C F \) are the three altitudes of \( \triangle A B C \). Show that \( \begin{array}{ll}\text { (i) Perimeter of triangle } A B C>2 A D & \text { (ii) Perimeter of triangle } A B C>2 B E \\ \text { (iii) Perimeter of triangle } A B C>2 C F & \text { (Perimeter of triangle } A B C)>2(A D+B E+C F)\end{array} \)

Dado un cilindro con un radio de 4 cm, ¿cuál debe ser la altura para que el volumen sea \( 64\pi \) cm³?