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Teorema geometrico dedicado a Marisol
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The Marisol Star Theorem
A dedication to our beloved lady Pepa Flores Marisol
Definition 1. Pentagons and stars.
Let P1, P2, P3, P4 and P5 be five arbitrary points in a plane. Then the pentagon P1P2P3P4P5 is defined as the union of the five line segments P1P2, P2P3, P3P4, P4P5 and P5P1. The respective star of this pentagon is defined as the union of the five line segments P1P3, P2P4, P3P5, P4P1 and P2P5.
Note that, by our definition, pentagons do not have to be "regular" or even "convex". A pentagon may look like a "star" if we label the five vertices in a particular way. For example, replace the original P1, P2, P3, P4 and P5 with P1, P3, P5, P2 and P4 respectively.
Definition 2. The inner pentagon of a pentagon (or star).
Let P1, P2, P3, P4 and P5 be an arbitrary pentagon (see the diagram). Let F4 be the point of interception of line segments P1P3 and P2P5; F5 be the point of interception of line segments P2P4 and P1P3; F1 be the point of interception of line segments P2P4 and P3P5; F3 be the point of interception of line segments P5P2 and P1P4; F2 be the point of interception of line segments P5P3 and P1P4 . Then we call the pentagon formed by F1, F2, F3, F4 and F5 the inner pentagon of the original pentagon (or star) formed by P1, P2, P3, P4 and P5.
Definition 3. The Marisol Star of a pentagon.
Let P1P2P3P4P5 be an arbitrary pentagon. Let M1 be the point of interception of P1F1 and P3P4, M2 be the point of interception of line segments P2F2 and P4P5; M3 be the point of interception of line segments P3F3 and P1P5; M4 be the point of interception of line segments P4F4 and P1P2; M5 be the point of interception of line segments P5F5 and P2P3. Then the star formed by M1, M2, M3, M4 and M5 is called the Marisol Star (the orange star in the diagram) of the original Pentagon formed by P1, P2, P3, P4 and P5.
Now we are ready to state the first theorem about the Marisol Stars:
Theorem 1. Let P1P2P3P4P5 be an arbitrary pentagon then the five vertices of the inner pentagon of the pentagon formed by its Marisol Star are all on the star formed by the original pentagon P1P2P3P4P5.
In other words, this theorem claims that P2P5, M3M5 and M2M4 meet in one and only one point Z1; P1P3, M1M4 and M3M5 meet in one and only one point Z2; P2P4, M2M5 and M1M4 meet in one and only one point Z3; P3P5, M1M3 and M2M5 meet in one and only one point Z4; P1P4, M1M3 and M2M4 meet in one and only one point Z5. Therefore, the Marisol star (the orange one, see the diagram) and the star (the black one, see the diagram) formed by line segments P1P3, P3P5, P5P2, P2P4 and P4P1 meet in five points Z1, Z2, Z3, Z4 and Z5. The inner pentagons (Z1Z2Z3Z4Z5 and F1F2F3F4F5) of the two stars (the orange Marisol Star and the black star in the diagram) meet each other in five and only five points and thus the two stars form a simple and beautiful structure.
The proof of this theorem uses the Menelaus Theorem extensively.
An even more interesting result is stated in the following:
Theorem 2. The Marisol Star (the pink star in the diagram) of the pentagon F1F2F3F4F5 (or the green star in the diagram) is the star formed by the five vertices of the inner pentagon of the Marisol Star (the orange star) of the original pentagon P1P2P3P4P5.
This relationship can repeat for infinite number of times as the Marisol Stars being scaled down in size. The interested readers should try to see what would happen to these two stars if we label the five points P1, P2, P3, P4 and P5 in a different order.
The author, now a PhD in Mathematics, discovered and proved the above theorems when he was 15 years old -- the year when he first saw the movie "Un Rayo de Luz". He is honored to name the star formed by M1, M2, M3, M4 and M5 in theorem 1, the Marisol Star, dedicated to a wonderful lady Pepa Flores / Marisol who has inspired him the most.
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Click here for the Marisol Star Diagram
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