A rectangle P is divided into smaller rectangles by segments parallel to its sides. We call a point a t-point if its a vertex of two small rectangles.

Question

A rectangle P is divided by segments parallel to its sides into smaller rectangles. We call a point a t-point if it is a vertex of exactly two such small rectangles. Prove that the number of t-points is even.

Answer 1

There are 3 kinds of points: t-point ($x$ points), points of the initial rectangle P (4 points) and points which are vertex of 4 rectangles($y$ points).

Suppose there are $n$ rectangles.

We calculate the total angle of these rectangles: $$2n\pi = 2x \frac{\pi}{2}+ 4 \frac{\pi}{2} +4y \frac{\pi}{2} $$ Then $$2n =x+2+2y$$ Hence, $x$ must be even number. Q.E.D

Answer 2

$t $ points should live on the outer, big rectangle. Now if such a point is one side then another one should live on the opposite side of the big regtangle. I think this is enough.

Unless by segments you mean not parallels lines that cut the initial rectangle into 2 points then $t$ points can be also an odd number.

Is the point A, the only t point?