What is the projective theorem?

If you need to edit answers or insert pictures, please click the title to enter the question details page. The so-called projection is orthographic projection. 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. [ 1]

Formula: As shown in the figure, in Rt△ABC, ∠ ABC = 90, and BD is the height on the hypotenuse AC, then the projective theorem is as follows:

( 1)(bd)^2=ad dc,(2)(ab)^2=ad AC ,(3)(bc)^2=cd ca .

Equal product formula (4)AB×BC=AC×BD (can be proved by "area method" or similar methods) (5) (AB) 2/(BC) 2 = AD/CD [1]

Proof of projective theorem of right triangle

Schematic diagram of projective theorem (geometric sketchpad)

(mainly calculated from the similarity ratio of triangles) 1,

In △BAD and △BCD, ∫ Abd+∠ CBD = 90, and ∠ CBD+∠ C = 90,

∴∠ABD=∠C，

∠∠BDA =∠BDC = 90。

∴△BAD∽△CBD

∴ AD/BD=BD/CD

Namely BD? = AD DC. The rest can be proved in the same way [1]