Properties of Triangles

If sin A,sin B,sin C are in A.P.
prove that the altitudes are in H.P .

1 Answers

2305

Shaswata Roy ·2014-01-24 07:05:30

Let's denote area by Î”.

\sin A,\sin B,\sin C \text{ are in AP}

\rightarrow 2R\sin A,2R\sin B,2R\sin C \text{ are in AP}

\rightarrow a,b,c \text{ are in AP}

\rightarrow \frac{1}{a},\frac{1}{b},\frac{1}{c} \text{ are in HP}

\rightarrow \frac{2\Delta}{a},\frac{2\Delta}{b},\frac{2\Delta}{c} \text{ are in HP}

\rightarrow h_a,h_b,h_c\text{ are in HP}

Where h_a,h_b,h_c are the altitudes of the triangle.

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