Show that the sum of the three altitudes of a triangle is less than the sum of three sides of the triangle.

Explanation

"Given: A triangle ABC in which `AD bot BC, BE bot AC and CF bot AB`.
To prove: `AD + BE + CF < AB + BC + CA or AD + BE + CF < Perimeter of triangle ABC `
Proof: As we know that from all the segments that can be drawn to a given line, from a point not lying on it, the perpendicular line segment is the shortest one.
`AD bot BC`
`AB > AD and AC > AD`
`AB + AC > 2AD` ---1
`BE bot AC`
`BC > BE and BA > BE `
`BC + BA > 2AB`--2
Also`, CF bot AB`
` AC > CF and BC > CF`
`AC + BC > 2CF` ---3
Adding 1,2 and 3, we get
`(AB + AC) + (AB + BC) + (AC + BC) > 2AD + 2BE + 2CF`
`2(AB + BC + CA) > 2(AD + BE + CF)`
` AB + BC + CA > AD + BE + CF `

Perimeter of `triangle ABC > AD + BE + CF` Hence proved"