Consider a generic triangle $▵ABC$, where $AB$ is the base, and $DE$ is the midsegment connecting the midpoints of sides $AC$ and $BC$, as illustrated in Figure 1.

The Triangle Midsegment Theorem is a fundamental principle in geometry that sheds light on the intricate properties of triangles and the fascinating interplay between their constituent parts. Essentially, this theorem posits that connecting the midpoints of the sides of any given triangle results in the formation of four smaller triangles, each possessing identical characteristics.

These smaller triangles, derived from the original triangle through the connection of its midpoints, exhibit remarkable symmetry and congruence. Notably, they share proportional sides and angles, mirroring the geometric intricacies inherent in the larger triangle. This revelation offers valuable insights into the underlying structure of triangles and unveils the hidden symmetries encoded within their configurations.

Furthermore, the Triangle Midsegment Theorem serves as a foundational tool in geometric analysis, providing a systematic approach to understanding the relationships between various components of a triangle. By dissecting the triangle into smaller, more manageable components, mathematicians and geometers can unravel the complexities of its geometry and derive meaningful conclusions about its properties and behavior.

In essence, the Triangle Midsegment Theorem stands as a testament to the elegance and sophistication of geometric principles, illuminating the inherent beauty and orderliness embedded within the realm of mathematics. Through its elucidation of the relationships between midpoints and triangles, this theorem enriches our understanding of geometry and empowers us to explore the intricate tapestry of shapes and forms that populate the mathematical landscape.

Let M denote any point situated on the midsegment $DE$ of triangle $▵ABC$. The midsegment theorem states that the midsegment of a triangle is parallel to its third side and is half its length. If the area of each triangle formed by connecting the midpoints of the sides of triangle $▵ABC$ is one-fourth the area of the original triangle, it indicates that these internally constructed triangles are precisely one-fourth the area of the larger triangle $▵ABC$. This observation aligns with the fundamental principles of the Triangle Midsegment Theorem.

Expanding on this concept, it can be asserted that when any point M lies on the midsegment $DE$, the triangles formed by connecting the midpoints of the sides of $▵ABC$ share an intrinsic relationship with the midsegment $DE$. This relationship underscores the theorem’s implications, illustrating that the areas of these internally constructed triangles are proportional to the area of the original triangle $▵ABC$.

Furthermore, this alignment with the Triangle Midsegment Theorem underscores the significance of the midsegment $DE$ in partitioning the triangle $▵ABC$ into smaller triangles with specific area relationships.

The summation of the areas of the triangles $▵AMC$ and $▵BMC$ is given by Eq. 5

Since

It can be demonstrated that irrespective of the size or configuration of a triangle, and regardless of the placement of any point along its midsegment, the relationship between the areas of the three resulting internal triangles adheres to the principles outlined in the Geometric Proof of the Pythagorean theorem. This means that the combined areas of the two smaller internal triangles, as defined by Equation 5, will always equal the area of the larger internal triangle, as defined by Equation 6. This fundamental relationship underscores the inherent geometric harmony within triangles, showcasing the timeless validity of the Pythagorean theorem across various geometrical contexts. Thus, regardless of the specific attributes of the triangle or the position of the point along its midsegment, this principle remains steadfast, providing a reliable framework for understanding and exploring geometric relationships within triangles.

For the case shown in Figure 2 where $M=D$, it holds true that

It is not unexpected that the connection between the internal triangle persists, upholding the assertion of Eq. 9. This enduring relationship underscores the stability and consistency of the mathematical framework being discussed. As we delve deeper into the analysis, we find that this connection serves as a foundational element, guiding our understanding and interpretation of the mathematical expressions at hand. Moreover, this ongoing correlation reaffirms the validity of the principles established earlier, lending further credence to the conclusions drawn from the mathematical model under scrutiny.

In the situation depicted in Figure 3, where M is not inside the $▵ABC$, the Pythagorean structure undergoes a transformation such that the Eq. 7 becomes the relationship shown in Eq. 10.

Certainly, the triangle $▵AMC$ has been meticulously enlarged to assert its dominance as the largest among the trio of triangles under consideration. This deliberate augmentation is executed with utmost precision, meticulously ensuring that the fundamental Pythagorean composition remains intact throughout the transformation process. By enlarging $▵AMC$ to its maximal extent within the given context, every effort is made to maintain the geometric integrity and inherent relationships that characterize the original configuration. This strategic enlargement serves to accentuate the significance and prominence of $▵AMC$ within the geometric framework, emphasizing its pivotal role in the analysis or application at hand. Through this deliberate action, the integrity of the underlying Pythagorean structure is preserved, while simultaneously amplifying the visual and analytical impact of $▵AMC$ within the broader geometric context.

Now, let’s delve into the intricate details of the general triangle $▵ABC$ as illustrated in Figure 4. Within this geometric framework, the sides $AB$, $AC$, and $BC$ symbolize the foundational elements of the triangle.

Expanding our exploration further, consider the formation of three distinct triangles: $▵APB$, $▵BPC$, and $▵APC$. These triangles emerge through the connection of a generic point $P(x,y,z)$ situated within three-dimensional space with the sides $AB$, $AC$, and $BC$ of the main triangle $▵ABC$.

In essence, by introducing this additional point P, we extend our analysis beyond the two-dimensional plane, allowing for a comprehensive examination of the geometric relationships within the spatial context. This expansion broadens the scope of our investigation, offering insights into the dynamic interplay between points and lines within a three-dimensional geometric framework.

Through the exploration of these supplementary triangles, we aim to uncover deeper insights into the structural properties and geometric configurations present within the general triangle $▵ABC$.

The altitude h of triangle $▵ABC$ and the distance l from its altitude to the y-axis are both established based on the triangle’s area, which is calculated utilizing the concise Heron’s formula, providing a streamlined method for determining these geometric properties with accuracy and efficiency.

Hence, h and l may be expressed in the following manner:

In order to illustrate the concept efficiently, we employ a computationally streamlined equation requiring only one square root operation. This equation facilitates the calculation of the area of a triangle in coordinate geometry. The formula, denoted as Eq. 17, operates as follows: it utilizes the coordinates of the vertices $(x_1,y_1,z_1)$, $(x_2,y_2,z_2)$, and $(x_3,y_3,z_3)$ to determine the area of the triangle situated within three-dimensional space.

In the subsequent iterations of our analysis, we employ Equation 17 to compute the areas of the triangles $▵APB$, $▵BPC$, and $▵APC$ individually. For the specific case of $▵APB$, let’s denote the coordinates of its vertices as follows: $x_1=0$, $y_1=0$, $z_1=0$, $x_2=a$, $y_2=0$, $z_2=0$, $x_3=x$, $y_3=y$, and $z_3=z$.

This setup allows us to precisely delineate the spatial configuration of $▵APB$ within the coordinate system. The vertex A is anchored at the origin $(0,0,0)$, while B is positioned at $(a,0,0)$ along the positive x-axis. The third vertex, P, holds the coordinates $(x,y,z)$, allowing for flexibility in the placement of this point within the spatial domain.

With these coordinate specifications in place, we proceed to apply Equation 17 to calculate the area of $▵APB$. This area computation involves the determination of the magnitude of the cross-product of the vectors $\overrightarrow{AB}$ and $\overrightarrow{AP}$. Through this method, we derive the precise numerical value representing the area enclosed by the vertices A, P, and B.

Similarly, we extend this computational procedure to ascertain the areas of $▵BPC$ and $▵APC$. Each triangle’s unique configuration, defined by the specified coordinates of its vertices, allows us to systematically evaluate its area utilizing the established mathematical framework.

Regarding the triangle $▵BPC$ where $x_1=l,y_1=h,z_1=0,x_2=a,y_2=0,z_2=0,x_3=x,y_3=y,z_3=z$ with

Concerning the triangle $▵APC$ where $x_1=l,y_1=h,z_1=0,x_2=0,y_2=0,z_2=0,x_3=x,y_3=y,z_3=z$

Utilizing the Pythagorean geometric proof concept within the context of the irregular tetrahedron $ABCP$ not only offers a profound insight into its structural intricacies but also reveals a plethora of mathematical solutions. The application of this concept leads to the derivation of three distinct solutions, each shedding light on different aspects of the tetrahedron’s geometry and relationships among its vertices.

Equation 23 serves as a crucial tool in elucidating these solutions, providing a mathematical framework through which the Pythagorean theorem can be applied to analyze the relationships between the lengths of the sides and diagonals of the tetrahedron. Through this equation, we can explore the various configurations and arrangements of the vertices A, B, C, and P, uncovering the intricate interplay between their spatial coordinates and distances.

Each solution derived from Equation 23 offers valuable insights into the geometric properties of the irregular tetrahedron $ABCP$. By leveraging the Pythagorean geometric proof concept, we can unravel the mysteries hidden within the complex geometry of tetrahedra.

By substituting the terms from equations 18, 19, and 22 into equation 23, we derive three respective implicit solutions for the hypothesized theorem, which are tailored specifically to tetrahedrons. We hereby denote these tetrahedrons as the Pythagorean Tetrahedron. These solutions encapsulate the geometric relationships and properties inherent to these specially defined tetrahedrons, thus providing a framework for understanding their structural characteristics and mathematical properties in the context of Pythagorean principles.

Expanding upon this, we can delve into the intricate interplay between the constituent elements of the Pythagorean Tetrahedron and how they contribute to its overall geometry and symmetry. Through rigorous analysis and mathematical manipulation, we elucidate the fundamental principles governing these tetrahedrons and establish their significance within the broader realm of geometric theory.

Furthermore, these implicit solutions offer insights into the spatial configurations and dimensional constraints of Pythagorean Tetrahedrons, shedding light on their unique structural attributes and potential applications in various mathematical and scientific disciplines.

It is imperative to clarify that the concepts elucidated in this discussion should not be conflated with the notion of the Trirectangular Tetrahedron or with De Gua’s theorem [5]. De Gua’s theorem posits a relationship within a tetrahedron that contains a right-angle corner akin to a corner on a cube. According to De Gua’s theorem, the sum of the squares of the areas of the three faces adjoining the right-angle corner equals the square of the area of the face opposite that corner. This theorem provides valuable insights into the geometric properties of such tetrahedra, shedding light on the intricate interplay between their constituent faces and angles.

In this section, we offer a visual exploration of the solution space provided by the implicit equations labeled as Eq.24,25, and 26, which directly relate to the conjectured theorem discussed in Section 3. Leveraging the computational capabilities of a computer algebra system (CAS), we present three-dimensional graphical representations that vividly depict the solutions to these equations. Through these visualizations, we aim to provide intuitive insights into the geometric relationships and patterns embedded within the theorem under consideration. We commence the visual exploration of resultant curvature by utilizing the classic 3-4-5 right triangle as our foundation. Through graphical representation of equations 24, 25, and 26, we unveil the emergence of three distinct bell-shaped tri-dimensional curvatures extending outward from the midsegments of triangle $▵ABC$. This depiction is vividly illustrated in Figure 5. Foreseen is the indefinite expansion of these bell-curvatures, suggesting their potential for perpetual growth. The void spaces within this geometric configuration encompass not only the internal triangle enclosed by the midsegments but also three supplementary regions generated by extending these midsegments in a manner that circumvents intersection with the vertices of the prototypical triangle. Figure 6, Figure 7, and Figure 8 provide visual representations of the projection of the resultant surface onto the $xy$-plane. In these figures, the original triangle is highlighted using bold black lines, while the contour levels are delineated by green lines for 3-4-5 right triangles, 3-3-3 isosceles triangles, and 3-4-6 scalene triangles. The conjectured Theorem 1, elucidated in Section 2, can be easily substantiated by examining the midsegments of these triangles. Expanding further, the projections offer a comprehensive view of how the resulting surface interacts with the $xy$-plane, showcasing the intricate relationships between the original triangle and its transformed counterparts. The use of bold black lines emphasizes the foundational structure of the original triangle, serving as a reference point for comparison. Meanwhile, the varying contour levels, depicted by green lines, highlight the distinct characteristics of each triangle type, facilitating a nuanced analysis of their geometric properties. By scrutinizing the midsegments within these projections, one can discern patterns and relationships that affirm the conjectured Theorem 1. The alignment and distribution of midsegments provide compelling evidence of the underlying geometrical principles at play, further reinforcing the validity of the theorem as proposed in Section 2. Overall, the visualizations presented in Figure 6, Figure 7, and Figure 8 offer valuable insights into the geometric transformations and relationships explored within this study. Thus, as posited, our demonstration has elucidated that the trio of inner triangles formed by joining a point on a midsegment with the vertices of the triangle adheres rigorously to the geometric validation of the Pythagorean theorem. This discovery not only reinforces the timeless relevance of the Pythagorean theorem but also sheds light on a deeper geometric connection. This connection, reminiscent of an expanded interpretation of the Pythagorean theorem, unveils a distinct spatial configuration defined by the intricate interplay of triangle areas within any given triangle. Moving forward, it becomes imperative for future endeavors to focus on the refinement and development of explicit formulas corresponding to the newly introduced concepts within our current framework. By pursuing this avenue of research, we can delve deeper into the mathematical intricacies and unveil further insights into the fundamental relationships that govern geometric structures. Such endeavors promise to enrich our understanding of geometry and pave the way for new avenues of exploration within the realm of mathematical theory and application.