Try this beautiful problem from the Pre-RMO, 2017 based on Altitudes of triangle.
Suppose the altitudes of a triangle are 10, 12 and 15, find its semi perimeter.
Altitudes
Triangle
Semi-perimeter
But try the problem first...
Answer: is \(\frac{60}{\sqrt{7}}\)
PRMO, 2017, Question 17
Geometry Vol I to IV by Hall and Stevens
First hint
\(h_a:h_b:h_c\)=10:12:15
or, a:b:c=\(\frac{1}{10} : \frac{1}{12} : \frac{1}{15}\)=6:5:4
or, (a,b,c)=(6k,5k,4k)
or, 2s=15k
Second Hint
\(\Delta=\sqrt{\frac{15k}{2}(\frac{15k}{2}-6k)(\frac{15k}{2}-5k)(\frac{15k}{2}-4k)}\)
or, \(\Delta=\frac{k^215\sqrt{7}}{4}\)
Final Step
\(h_{10}=10 =\frac{2k^2\sqrt{7}\frac{15}{4}}{6k}\)
or, k=\(\frac{8}{\sqrt{7}}\)
or, s=\(\frac{60}{\sqrt{7}}\)
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