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## QuestionMathsClass 11

In a triangle $$ABC,$$ if $$\sin{A},\,\sin{B},\,\sin{C}$$ are in A.P then
(A) altitudes are in A.P
(B) altitudes are in H.P
(C) medians are in G.P
(D) medians are in A.P

$$\sin{A},\,\sin{B},\,\sin{C}$$ are in A.P

$$\Rightarrow a,\,b,\,c$$ are in A.P

We know that Area of a triangle $$\Delta=\dfrac{1}{2}a{h}_{1}=\dfrac{1}{2}b{h}_{2}=\dfrac{1}{2}c{h}_{3}$$

$$2\Delta=a{h}_{1}=b{h}_{2}=c{h}_{3}$$

$$\therefore {h}_{1}:{h}_{2}:{h}_{3}$$

$$=\dfrac{2\Delta}{a}=\dfrac{2\Delta}{b}=\dfrac{2\Delta}{c}$$

$$=\dfrac{1}{a}=\dfrac{1}{b}=\dfrac{1}{c}$$

$$\Rightarrow a,\,b,\,c$$ are in A.P

$$\Rightarrow {h}_{1},\,{h}_{2},\,{h}_{3}$$ are in H.P
Hence proved