Convex Sets and Triangles - Mathematical Proof

Q1)
P
C
E
A D B
K T
Consider triangle ∆ABC shown above.
Let P be the half plane of line AB in which point C lies, let K be the half plane of line BC in which point A
lies, and let L be the half plane of line CA in which point B lies. By definition, the interior domain of the
triangle ∆ABC is the intersection P ∩ K ∩ T.
Using Hilbert’s Proposition, there exists a point D lying between A and B. using the same reasoning,
there exists a point E lying between C and D. claiming that E ∈ P ∩ K ∩ T, which is the interior of triangle
∆ABC. Indeed, point E lies in the same half plane P of line AB as point C. Indeed, the segment EC lies
entire in P since D ∗ E ∗ C. The points B, D and E lie in the same half plane T of line CA. Indeed, because
of A ∗ D ∗ B and D ∗ E ∗ C, segments DB and DE lie entirely in T.
Similarly, since segments AD and DE lie in the half plane K, we see that points A, D and E lie in the same
half plane K of line BC.
Q2)
Let assume that S is a convex having the two points A and B being in S
Lemma
If S is convex, T is convex, and then S ∩ T is convex
Let π be a half plane bounded by the line l
Let A and b be two points in the planeπ.
For any point P in the plane such that P≠ B the point (1 – t)P + tQ is in T
Note that A and B are on the same side of the linel.
Let T be an element of the line segment AB

If A = T, then T is in half plane π
If B = T, then T is in half plane π
Thus every point T is an element of segment AB inπ.
The π is a convex set
Q3.
The interior of triangle is always a convex set
Proof:
Denote the triangle as ∆, and the interior of the boundary of ∆ as int(∆)
From boundary of polygon is Jordan curve, it follows that the boundary of ∆ is equal to imag of a Jordan
curve, so int( ∆) is well defined.
Denote the vertices of ∆ as A1, A2, A3 for i ∈ {1 ,2 , 3 }, put j = 1mod3 +1, k = (i + 1)mod3 +1, and:
Ui = {Ai +st(Aj – Ai)+(1 – s)t(Ak – Ai): s∈(0..1), t∈ R >0 }
Suppose that ¿Ai is an integral in ∆ , it follows from definition of polygon that ¿Ai cannot be zero or
straight. Then ¿Ai is larger than a straight angle, which is impossible by sum angles of triangle equals two
right angles.
It follows that ¿Ai is convex
From characterization of interior of triangle, it follows that
Int(∆) = ¿ i=1 ¿ 3U i
From interior of convex angle is convex set, it follows for i∈ { 1 , 2, 3 } that Ui is a convex set.