Symmetric sum of altitudes

Geometry Level 3

Triangle $ABC$ has area 15 and perimeter 20. Furthermore, the product of the 3 side lengths is 255. If the three altitudes of the triangle have lengths $d, e$ , and $f$ , then the value of $de+ef+fd$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . What is $m+n$ ?

The answer is 1217.

1 solution

Shaun Leong
May 12, 2016

$\frac12ad=\frac12be=\frac12cf=15$ $d=\frac{30}{a}, e=\frac{30}{b}, f=\frac{30}{c}$ $de+ef+fd=900 (\frac {1}{ab}+ \frac {1}{bc}+ \frac {1}{ca})$ $=\frac{900(a+b+c)}{abc}$ $=\frac{900*20}{255}$ $=\boxed{\dfrac{1200}{17}}$

Same approach.

Niranjan Khanderia - 5 years ago

Multiplied by 17 instead of dividing

Aryan Dutt - 1 year, 6 months ago

What is a b and c here could you show it in a figure?

Lucky Narasimhan - 3 years, 7 months ago

Very good.👍🏻

Rudrayan Kundu - 2 years, 10 months ago

This solution is wrong 255 is the product of altitudes. i.e. d.e.f. =255. Right answer is 341.

Kamal Suthar - 4 months, 1 week ago

Symmetric sum of altitudes

The answer is 1217.

1 solution

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