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The problem:

Three poles standing at the points $A$, $B$ and $C$ subtend angles $\alpha$, $\beta$ and $\gamma$ respectively, at the circumcenter of $\Delta ABC$.If the heights of these poles are in arithmetic progression; then show that $\cot \alpha$, $\cot \beta$ and $\cot \gamma $ are in harmonic progression.

Now, what I could not understand is subtending of the angle part,precisely how a point subtends angle at another point? So, what I am looking for a proper explanation of the problem statement with a figure, since it's troubling me from sometime.

PS: I am not looking for the solution (as of now) or any hint regarding the solution, just a clear explanation will be appreciated.


My solution using Moron's interpretation,

Let $a$ $b$ and $c$ are the length of three sides of the poles and $O$ be the circumcenter then, $$ \cot \alpha = \frac{OA}{a}$$ $$ \cot \beta = \frac{OB}{b}$$

$$ \cot \gamma = \frac{OC}{c}$$

As $O$ is the circumcenter,$OA = OB = OC = k $(say)

Again, $a$ $b$ and $c$ are in arithmetic progression, hence

$$2 \cdot \frac{k}{\cot \alpha} = \frac{k}{\cot \beta} + \frac{k}{\cot \gamma}$$

Canceling $k$ from both sides,

$$2 \cdot \frac{1}{\cot \alpha} = \frac{1}{\cot \beta} + \frac{1}{\cot \gamma}$$

Hence, $\cot \alpha$, $\cot \beta$ and $\cot \gamma $ are in harmonic progression. (QED)

2 Answers 2

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My guess is the poles are of (different) heights $h_A$, $h_B$, $h_C$ and the angle is from the foot of pole to the circumcentre of $\triangle ABC$ to the top of the pole.

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    Aha .. Do you meant the $3D$ angle?2010-12-03
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    @Deb: Yes 3D, seems to make sense when talking about poles and angle subtended.2010-12-03
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    Well,your guess is bang on! I solved the problem using your interpretation. :)2010-12-04
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    @Deb: Glad that worked :-)2010-12-04
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    Added my solution,just let me know if anything seems wrong to you :-)2010-12-04
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    @Deb: Yes, there is something wrong. Your question states "angles of poles are in arithmetic progression", while in the solution you are assuming that the heights of the poles are in arithmetic progression.2010-12-04
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    OOPs the actual problem is about the height of the poles :-P2010-12-04
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    I have fixed it now,but anyways angles in A.P also seems interesting!2010-12-04
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    @Deb: Yes angles and cot of angles in progression would be an interesting trigonometry problem. I am guessing it will only have trivial solutions.2010-12-04
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    In that case I guess it's not in H.P.2010-12-04
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    Umm does it even follows any know progression? I doubt!2010-12-04
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I would read it that the poles have a diameter>0. The pole at A has diameter so the angle seen from the circumcenter is $\alpha$. See if that works.

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    I think Moron's interpretation and mine will lead to the same answer. They just work in orthogonal axes.2010-12-03