What is a Tetrahedron?

A type of triangular pyramid, a geometric body, consisting of four triangles. There is one vertex when the bottom surface is fixed, and four vertices when the bottom surface is not fixed. (A regular pyramid is not the same as a regular tetrahedron. A regular tetrahedron must be a regular triangle on each face.)

What is a triangular pyramid

Geometry,

Helen Qin Jiushao Volume Formula

A volume formula is known for the length of a triangular pyramid.

Any triangular pyramid or tetrahedron, whose edges are a, b, c, d, e, f, where a and d, b and e, c and f are opposite sides of each other, then there is a volume of a triangular pyramid (tetrahedron) The formula is ^[3] :

Inscribed ball heart

The inscribed sphere center of the regular triangular pyramid is 1/4 of the bottom surface from the line connecting the vertex and the center of gravity of the bottom surface.

Relevant calculations: Because the bottom of the regular triangular pyramid is a regular triangle, the high line is located at any vertex and the midpoint of the bottom edge, and the three lines are combined. Also, knowing the length of the side of the regular triangular pyramid, you can calculate the length of the line where the center of the circle is located (that is, the line connecting the apex and the center of gravity of the bottom surface) according to the Pythagorean theorem. ^[3]

The projections of a general triangular pyramid inscribed sphere center on the four faces coincide with the center of gravity of the four faces, and the position of the sphere center can be determined accordingly.

Circumcenter

The circumscribed sphere center of the regular triangular pyramid is 1/4 of the bottom surface from the line connecting the vertex and the center of gravity of the bottom surface.

Related calculations: Calculate the length of the line where the center of the circle (the line connecting the vertex and the center of gravity of the bottom surface) is the same as calculating the center of the inscribed sphere.

The projection of a general triangular pyramid tangent sphere center on the four faces coincides with the perimeter of the four faces, and the position of the center of the sphere can be determined accordingly.

Where R is the radius of the circumscribed sphere, a, A, B are shown in the figure,

It is the dihedral angle of the plane where A and B are located.

If the dihedral angle is 90 °, that is, when the two sides are perpendicular, the formula is simplified as

Tangent sphere center

The center of the tangent of the triangle of the regular pyramid is 1/4 from the bottom of the line connecting the vertex and the center of gravity of the bottom (the triangles of the regular triangle are coincident)

The projection of the spherical center of a general triangular pyramid and four edges on the four faces coincides with the inner centers of the four faces, and the position of the spherical center can be determined accordingly.

Projection of the vertex of a triangular pyramid and the "heart" of the triangle on the bottom

A triangular pyramid P-ABC is set, and the projection of P on the plane ABC is O. Now when discussing what conditions the triangular pyramid meets, O is the outer center, inner center, paracenter, center of gravity, and vertical center of triangle ABC (triangle five centers). ^[4]

Outside heart

If O is the circumcenter of ABC, then OA = OB = OC. Since OP plane ABC (definition of projective), OPOA, OPOB, OPOC. Pythagorean theorem gives PA = PB = PC. In addition, tanPAO = OP / OA, tanPBO = OP / OB, and tanPCO = OP / OC, so we can know that PAO = PBO = PCO.

In summary, the following theorems can be obtained:

1. When the three side edges of a triangular pyramid are equal, the projection of the vertex on the bottom surface is the outer center of the bottom triangle.

2. When the three side edges of the triangular pyramid are equal to the angle formed by the bottom surface, the projection of the vertex on the bottom surface is the outer center of the bottom triangle. ^[4]

heart

If O is the inside of ABC, the distance from O to the three sides is equal, and O is within ABC. Let the vertical lines from O to BC, AC, and AB be OD, OE, and OF, respectively, then OD = OE = OF. PD = PE = PF is obtained from the Pythagorean theorem. And tanPDO = OP / OD, tanPEO = OP / OE, tanPFO = OP / OF, so PDO = PEO = PFO. And from the three perpendicular theorem, we can know that PDBC, PEAC, PFAB, that is, PDO, PEO, PFO are dihedral angles P-BC-A, P-AC-B, P-AB-C Flat angle.

In summary, the following theorems can be obtained:

1. When the vertices of the triangular pyramid have the same distance from the three sides of the triangle to the bottom, and the projection of the vertex on the bottom is inside the triangle on the bottom, then the projection is inner.

2. When the dihedral angle formed by each side of the triangular pyramid and the bottom surface is equal, and the projection of the vertex on the bottom surface is inside the triangle of the bottom surface, then the projection is the heart.

Side by side

Because the properties of the inward and the inward are the same, they are points with equal distance to the three sides of the triangle. It's just that the heart is inside the triangle and the side is outside the triangle. So the discussion is the same as the heart, the difference lies in the positional relationship between O and ABC. So we get the following theorem directly:

1. When the vertices of the triangular pyramid have the same distance from the three sides of the triangle to the bottom, and the projection of the vertex on the bottom is outside the triangle on the bottom, the projection is paracentric.

2. When the dihedral angle formed by each side of the triangular pyramid and the bottom surface is equal, and the projection of the vertex on the bottom surface is outside the triangle on the bottom surface, then the projection is paracentric.

Cherish

If O is the vertical center of ABC, there are OABC, OBAC, OCAB. Because O is a projective of P, we can know PABC, PBAC, PCAB from the triple perpendicular theorem. In terms of generalization, PA BC can be associated with PAplane PBC, and according to the theorem of line and plane vertical determination, the conditions of PAplane PBC are PAPB, PAPC. Similarly, PBPA, PBPC; PCPA, PCPB. That is, PA, PB, and PC are perpendicular to each other. ^[4]

In summary, the following theorems can be obtained:

1. When the three side edges of a triangular pyramid are perpendicular to each other (or each side edge is perpendicular to the opposite side), the projection of the vertex on the bottom surface is the perpendicular of the bottom triangle.

2. When a triangular pyramid has two side edges that are perpendicular to the corresponding opposite side, the third group of side edges is also perpendicular to the opposite side, and the projection of the vertex on the bottom surface is the perpendicular of the bottom triangle.

Center of gravity

theorem:

1. When the sum of the square of any side of a triangular pyramid and the square of its opposite edge is a fixed value, the projection of the vertex of the triangular pyramid on the bottom surface is the center of gravity of the bottom surface.