Given two points on a circumference of radius $R$, $P_0$ and $P_1$ subtended by an angle $\theta$ at the center of the circumference, what is the angle at which a generic point $P_m$ inside the circle 'sees' the two points $P_0$ and $P_1$?
A simple question about angles on a circumference
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0@Riccardo.Alestra . I think we *all* understood that from the beginning, and $\,P_m\,$ is **inside** the circle, as you wrote. – 2012-09-12
1 Answers
With $P_0, P_1$ as given and $O$ as center of the circle, extend the line $P_0P_m$ until it meets the circle in $Q$. Let us assume that $P_m$ is on the same side of $P_0P_1$ as $O$. Then by the inscribed angle theorem we have $\theta = \angle P_0OP_1 = 2\angle P_0QP_1$. In triangle $P_1QP_m$, we have $\angle QP_1P_m + \angle P_mQP_1+\angle P_1P_mQ=2\pi$ and we also have $\angle P_0P_mP_1+\angle P_1P_mQ=\angle P_0P_mQ=2\pi$, hence $\angle P_0P_mP_1=\angle QP_1P_m + \angle P_mQP_1> \angle P_mQP_1= \angle P_0QP_1 = \frac12 \theta$.
There is a slight subtlety involved when $P_m$ is on the other side of $P_0P_1$ and also beware of orientation. All in all you will find that the range $\frac12\theta<\angle P_0P_mP_1<\frac12\theta+\pi$ is possible.
Here's a proof variant with coordinate calculations: Without loss of generality, $R=1$, the center is $(0,0)$, $P_0=(c, -s)$, $P_1=(c,s)$ with $0\le c=\cos\frac\theta2<1$, $0 and $P_m=(x,y)$ with $r^2:=x^2+y^2<1$. We can find the sine of $\alpha:=\angle P_0P_mP_1$ by calculating the $z$ coordinate of the crossproduct $\overrightarrow{P_mP_0}\times \overrightarrow{P_mP_1}$ and dividing by the lengths of the factors. The $z$ coordinate of the cross product is $(c-x)(s-y)-(-s-y)(c-x)=2s(c-x) $. Hence $\sin\alpha = \frac{2s(c-x)}{|P_mP_0||P_mP_1|}.$ We can also find the cosine of $\alpha$ via the scalar product: $\cos\alpha=\frac{\overrightarrow{P_mP_0}\cdot \overrightarrow{P_mP_1}}{|P_mP_0||P_mP_1|}= \frac{(c-x)^2+y^2-s^2}{|P_mP_0||P_mP_1|}$ The lngths in the denominators are a bit clumsy, but we can ignore them if we only need to test for the sign of $\sin\alpha$ and $\cos\alpha$ and compute the tangent of $\alpha$: $\tan\alpha = \frac{\sin\alpha}{\cos\alpha}=\frac{2s(c-x)}{(c-x)^2+y^2-s^2}.$ First note that $\sin\theta=2cs$ and $\cos\theta=c^2-s^2$ by the double angle formulas, hence $\tan\theta=\frac{2cs}{c^2-s^2}$. I claim that one of the follwoing cases holds:
- $\cos\alpha>0, \sin\alpha>0, \tan\alpha>\tan\theta$
- $\cos\alpha\le 0, \sin\alpha\ge 0$
- $\cos\alpha<0, \sin\alpha<0, \tan\alpha<\tan\theta$
For each of these cases you can manipulate the expressions given for the trigonometric functions to verify the inequality in the given situation. It is a bit lengthy, and I'm a bit tired, though
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0$I$ argued in classic geometry. I'll add an answer purely using cartesian coordinates. – 2012-09-13