6. $$ A D, B E $$ and $$ C F $$ are the three altitudes of $$ riangle A B C $$. Show that $$ \begin{array}{ll} ext { (i) Perimeter of triangle } A B C2 A D & ext { (ii) Perimeter of triangle } A B C2 B E \ ext { (iii) Perimeter of triangle } A B C2 C F & ext { (Perimeter of triangle } A B C)2(A D+B E+C F)\end{array} $$

Question

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

Frazier Wells · Accepted Answer

To prove the inequalities for triangle $$ ABC $$ with altitudes $$ AD, BE, $$ and $$ CF $$: 1. **Perimeter $$ P > 2AD $$**: Using the area and triangle inequality, we show that the sum of two sides is greater than the third side, leading to $$ P > 2AD $$. 2. **Perimeter $$ P > 2BE $$**: Similar to above, using the area and triangle inequality, we derive $$ P > 2BE $$. 3. **Perimeter $$ P > 2CF $$**: Again, applying the area and triangle inequality, we find $$ P > 2CF $$. 4. **Combining the Results**: Adding the three inequalities, we conclude that: $$ P > 2(AD + BE + CF) $$ This completes the proof of all parts.

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