I have recently published a note in the International Journal of Computer Discovered Mathematics where I prove a conjecture by Van Khea. The result is as follows:
Theorem (Van Khea): Let $ABC$ be a triangle, and $P$ any point on the plane of $ABC$. Let $X$, $Y$, and $Z$ be arbitrary points on sides $BC$, $AC$, and $AB$, respectively. Let $D$ be the reflection of $P$ around $X$. Similarly, define $E$ and $F$. Denote $U$, $V$, and $W$ as the midpoints of sides $BC$, $AC$, and $AB$, respectively. Let $D'$ be the reflection of $D$ around $U$. Similarly, define $E'$ and $F'$. Then:
$$[DEF] + [D'E'F'] = [ABC],$$
where $[]$ denotes areas.
For some time, I've been contemplating the idea that there might be an analogous/extended version of this theorem related to tetrahedra. Today, I conducted some experiments using Geogebra and a volume calculator that utilizes the matrix formula. Based on my findings, I arrived at the following conjecture:
Conjecture: Let $ABCD$ $(T_1)$ be any tetrahedron with vertices lying on a sphere, and $P$ be an arbitrary point in space. Denote $G_{ABC}$ as the centroid of the face $ABC$, and $Q_{ABC}$ as an arbitrary point inside $ABC$ and lying in the same plane. Let $P'_{ABC}$ be the reflection of $P$ with respect to $Q_{ABC}$. Similarly, define $P'_{ABD}$, $P'_{ACD}$, and $P'_{BCD}$. Let $T_2$ be the tetrahedron defined by these four points. Now, denote $P''_{ABC}$ as the reflection of $P'_{ABC}$ with respect to $G_{ABC}$. Analogously, define $P''_{ABD}$, $P''_{ACD}$, and $P''_{BCD}$, and let $T_3$ be the tetrahedron defined by these points. Then: $$V_2 + V_3 = V_1,$$
where $V_1$, $V_2$ and $V_3$ denote the volumes of $T_1$, $T_2$ and $T_3$, respectively.
The proof I have in mind uses analytic geometry and the matrix formula. I know how to find the coordinates of the centroids and the reflected points, but I have no idea how to parameterize the arbitrary points $Q$ on the faces of the tetrahedron. Could someone confirm whether this conjecture is true?
Unfortunately, the conjecture is false for tetrahedra. $$A (8.81677261342614, -5.95109812523592, 0.00000000000000)\\ B (-0.868026162586332, 6.82346541589857, 0.00000000000000)\\ C (6.99038460398954, 4.45229870144277, 1.56030015466736)\\ D (0.987740774325822, 1.98714940900149, 4.02997763720348)$$ Volume: 58.01800458
$$P'ABC (8.81867941456366, 15.3602994734030, 1.05673420386347)\\ P'ABD (11.5702309614861, 5.30685185335826, 1.70376656686520)\\ P'ACD (10.1012477803281, 10.7579977259050, 5.40704749520762)\\ P'BCD (6.18230217961132, 14.1245366398754, 5.51507484080838)$$ Volume: 20.37474752
$$P''ABC (1.14074128865590, -11.8105221453327, -0.0165341007519011)\\ P''ABD (-5.61257281137571, -3.40050738691550, 0.982885191270448)\\ P''ACD (1.09535088083288, -10.4324310690995, -1.68019563396039)\\ P''BCD (-1.44223603579196, -5.28259428898019, -1.78822297956115)$$ Volume: 3.02160658