deletions | additions       diff --git a/Observe_the_angles_angle_PQ__.tex b/Observe_the_angles_angle_PQ__.tex  index d8c080b..5645d0b 100644  --- a/Observe_the_angles_angle_PQ__.tex  +++ b/Observe_the_angles_angle_PQ__.tex 
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\therefore \alpha := \max \left( PQ'A, PQ'B \right) &\geq \frac{\pi}{2}  \end{align}  As such $\alpha$ must be the unique largest internal angle in the triangle $PQ'B$. Consequently the side opposite $\alpha$ -- $PB$ -- is the unique hypotenuse of the triangle. Notably, this means As such  $|PB| > |PQ'|$.   However $P$ and $B$ are both elements of the convex hull, so by the convexity property we can draw |PQ'|$,  a non-intersecting line segment between them. contradiction.