Three conclusions of projective theorem

Three conclusion of that projective theorem are as follows:

Projection theorem of right triangle (also called Euclid theorem): In a right triangle, the height on the hypotenuse is the proportional average of the projections of two right angles on the hypotenuse. Each right-angled edge is the median of the projection of this right-angled edge on the hypotenuse and the proportion of the hypotenuse.

In the formula Rt△ABC, ∠ BAC = 90, and AD is the height on the hypotenuse BC, then the projective theorem is as follows: (1) (ad) 2; =BD DC，(ab)^2； = BD BC, (3) (AC) 2; =CD BC. Equal product formula (4)ABXAC=BCXAD (proof of usable area)

Area projection theorem: "The projection area of a plane figure is equal to the area s of the projected figure multiplied by the cosine of the included angle between the plane where the figure is located and the projection plane." COSθ=S projection /S primitive (the areas of planar polygons and their projections are s primitive and s projection, respectively, and the dihedral angle formed by their planes is θ).

Proof idea: Because the projection is to scale the length of the original figure (the height in the triangle) and the width is unchanged, and because the area ratio of the plane polygon = the square ratio of the side length. So it is the ratio of the length of the figure (called the height in the triangle). Then this ratio should be the cosine of the angle formed by the plane.