If there is a point O in the plane of △ ABC, and OA * ob = ob * OC = OC * OA, then point O is the () center of △ ABC

If there is a point O in the plane of △ ABC, and OA * ob = ob * OC = OC * OA, then point O is the () center of △ ABC

OA*OB=OB*OC
OB*(OA-OC)=0
OB*CA=0 BO⊥CA
In the same way
CO⊥BA
O is the vertical center of △ ABC

If the side edges OA, OB and OC of a triangular pyramid o-abc are perpendicular and the side areas are 6, 4 and 3 respectively, the volume of a triangular pyramid is? Please explain

OA*OB=12,OC*OA=8,OB*OC=6
Volume = 1 / 3 * 1 / 2oa * ob * OC = 1 / 6 * 24 = 4

In tetrahedral o-abc, G and H are the barycenters of △ ABC and △ OBC respectively. Let OA = a, OB = B and OC = C. prove og = 1 / 3 (a + B + C)

(no picture, draw a sketch to understand)
Through the centrosymmetric transformation of point G, a tetrahedron o-abc which is centrosymmetric with respect to point G is obtained
O '- ABC, we know that the hexahedron o'oabc is parallelepiped
So the value of vector a + B + C is vector OO '
Next, let the center of gravity of o-abc be K, extend og to K ', so that Ko = OK',
Extend Oh to L
It can be proved that og = GG '= 2kg, so og = (2 / 6) * OO' = 1 / 3 (a + B + C)