Three vertical line theorems

Three vertical theorems: refers to a straight line in a plane. If it is perpendicular to the projection of a diagonal line passing through this plane on this plane, then it is also perpendicular to this diagonal line.

Proof and application of 1 and triple vertical theorem

The three perpendicular theorem determines that the oblique line is perpendicular to the straight line in the plane through the vertical relationship between the projection of the oblique line in the plane and the straight line in the plane. Because this theorem involves three straight lines (PA⊥ Plane α, PB⊥a, AB⊥a) perpendicular to the known straight lines in the plane, it is called the three perpendicular theorem.

2. Inverse theorem of three perpendicular lines theorem

If a straight line in a plane is perpendicular to the diagonal line passing through the plane, it is also perpendicular to the projection of the diagonal line in the plane.

Proof of the theorem of three vertical lines;

(1) Vertical line (plane problem)? Line oblique vertical (space problem).

(2) Proof methods of straightness: definition method, straightness judgment theorem; Three vertical theorems. The key to the application of the three perpendicular theorem is to find the perpendicular of the plane (datum). As for projection, it is determined by vertical feet and inclined feet, so it is the second type.

(3) The three vertical theorems describe the vertical relationship among PO (diagonal), AO (projection) and A (straight line). The key is to find the direction perpendicular to the plane (datum). As for projection, it is determined by vertical feet and inclined feet, so it is the second type.

(4) Lines A and PO may or may not intersect. Tri-perpendicularity theorem and inverse theorem are one of the most useful tools to solve dihedral angle, and also the common tools to prove the perpendicularity of straight lines.

(5) The essence of the Three Verticality Theorem is the judgment theorem that a diagonal line in a plane is perpendicular to a straight line in a plane.

(6) It can be used to solve the problems such as the angle formed by straight lines in different planes and the plane angle of dihedral angles. That is, the first alignment plane (datum plane) and the second alignment projection line are vertical, then A and B form a straight line and a diagonal line on the plane. Thirdly, it is proved that the projective straight line is perpendicular to the straight line A, thus the conclusion that A is perpendicular to B is drawn. ..