Show that the sum of three altitudes of a triangle is less than the sum of the three sides of the triangle.
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The sum of three altitudes of a triangle is less than the sum of the three sides of the triangle.
Explanation:
In ΔABC, AD, BE and CF are the medians to the sides BC, AC and AB respectively.
Therefore, we know that, the sum of any two sides of triangle is greater than twice the median bisecting the third side.
AB + AC > 2 AD
AB + BC > 2 BE
BC + AC > 2 CE
Adding these three equations, we get
AB + AC + AB + BC + BC + AC > 2 AD + 2 BE + 2 CF
⇒ 2AB + 2BC + 2AC > 2AD + 2BE + 2CF
⇒ 2 (AB + BC + AC) > 2(AD + BE + CF)
⇒ AB + BC + AC > AD + BE + CF
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