On a plane , there are $2n+1$ points where no three points are co-linear. Show that for any point $P$ which is one of the points, the number of triangles the interior of which $P$ lies in is always even.
My attempt : an easy observation is that for any fours points, the number of triangles $P$ lies in is either 0 or 2 & Conclusion holds for $n=2$. But I am not sure this is sufficient to give the argument for all $n$ in general.
Let the Points be $P_0,P_1,P_2,P_3,\cdots,P_{2n}$ & let us consider Point $P_0$ , leaving out the other $2n$ Points.
Put the $2n$ Points where-ever those are supposed to be.
Put $P_0$ in our control , temporarily : Put $P_0$ very far out.
When $P_0$ is very far out , it is outside all the triangles formed with the other $2n$ Points.
Let $C$ be the Count of the triangles inside which $P_0$ is [[ this $C$ is the number we are considering ]]
The Conclusion holds because $C=0$ is EVEN.
Now , gently & continuously move $P_0$ towards the Original Position ( which is not in our control )
We might have to cross some line Segments to achieve that.
Let us cross the line Segments 1 at a time.
Let $P_xP_y$ be the Current line Segment we are crossing.
This Segment forms triangles with all other $2n-2$ Points.
That gives us $2n-2=EVEN$ triangles with the Segment $P_xP_y$.
We can see that every such triangle is either on one side or on the other side of Segment $P_xP_y$.
Hence , the Count $C$ will reduce by the number of triangles on one side ( say $A$ ) & then increase by the number of triangles on the other side ( say $B$ ) where we have $A+B=2n-2=EVEN$.
Over-all Change in $C$ is given by $-A+B$ which , being Equal to $A+B-2A$ , is also EVEN.
Hence $C$ remains EVEN when-ever $P_0$ is crossing Segments.
The Conclusion holds at every Crossing , because $C$ is EVEN.
When $P_0$ eventually reaches Destination Position , it will still be EVEN.
The Conclusion holds because $C$ is EVEN when we terminate the movement after reaching the Destination Position.
CRUX : Basically the Parity is INVARIANT here.
DONE !
ADDENDUM :
When crossing the first Segment , then $C$ will change from $0$ to $2n-2$ , because $A=0$ & $B=2n-2$ , hence Change is $[-0]+[2n-2]$
After that , the further Changes will not be like this , but over-all Change will be EVEN.
Pictorial View :
When $P_0$ moves across Segments , then $C$ will change.
When considering Blue Segment , we can see that $C$ will exclude triangles on one side & include triangles on the other side. Change will be EVEN.