11ans to 1st one is 6.
Let ABC be any triangle with altitudes h1,h2,h3 and inradius r, then find the minimum value of
(h1+r)/(h1-r) +(h2+r)/(h2-r) +(h3+r)/(h3-r) =
If A,B and C are three non-collinear points in a plane, the area of the greatest equilateral triangle, which can be drawn to circumscribe triangle ABC is =
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3 Answers
21we know h_{1}=\frac{2\Delta }{a} h_{2}=\frac{2\Delta }{b} h_{3}=\frac{2\Delta }{c}
so given exp is =\frac{2\Delta /a +\Delta/s }{2\Delta /a -\Delta/s }+\frac{2\Delta /b +\Delta/s }{2\Delta /b -\Delta/s }+\frac{2\Delta /c +\Delta/s }{2\Delta /c -\Delta/s }
=\frac{2s+a}{2s-a}+\frac{2s+b}{2s-b}+\frac{2s+c}{2s-c}
=\frac{4s}{2s-a}+\frac{4s}{2s-b}+\frac{4s}{2s-c}-3 now
apply AM≥HM for \frac{4s}{2s-a}+\frac{4s}{2s-b}+\frac{4s}{2s-c}
3\left[\frac{1}{3}\left\{\frac{4s}{2s-a}+\frac{4s}{2s-b}+\frac{4s}{2s-c}\right\} \right]-3\geq 3\left(\frac{3}{\frac{2s-a}{4s}+\frac{2s-b}{4s}+\frac{2s-c}{4s}} \right)-3
\frac{4s}{2s-a}+\frac{4s}{2s-b}+\frac{4s}{2s-c}\right-3\geq 3.3-3
\geq 6