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- Apr 14, 2013

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On a unit circle $K$ there are twelve points $P_1, \ldots , P_{12}$ in that order and each two successive points have the same distance (as the number $1$ to $12$ on a clock). Let $g$ be the line that passes through $P_1$ and $P_8$ and let $h$ be the line that passes through $P_3$ and $P_{11}$. Let $S$ be the intersection of $g$ and $h$.

1. Show that the triangles $P_{11}P_8S$ and $P_1P_3S$ are similar.

2. Calculate the two acute angles that are formed by $g$ and $h$.

3. Calculate the side lengths of the triangle $P_1P_3S$.

For question 1 we have that at both triangles the angle of $S$ are similar, aren't they? What can we say about the other angles? the arcs are not equal.