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## 'Altitude Inequalities' printed from http://nrich.maths.org/

A triangle $T$ has an area of $1$cm$^2$. Let $M$ be the product of the perimeter of $T$ and the sum of the three altitudes of $T$. Which of the following statements is false?

A. There are (or there exist) triangles $T$ for which $M> 1000$

B. $M> 6$ for all triangles $T$

C. There are triangles $T$ for which $M=18$

D. $M> 16$ for all right-angled triangles

E. There are triangles $T$ for which $M< 12$

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*This problem is taken from the UKMT Mathematical Challenges.*

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