What is the projective theorem?

Projection Theorem, also known as Euclid Theorem: In a right triangle, the height on the hypotenuse is the median term of the ratio of the projection of two right-angled sides on the hypotenuse, and each right-angled side is the median term of the ratio of the projection of this right-angled side on the hypotenuse to the hypotenuse. Projective theorem is an important theorem in mathematical graphic calculation.

At Rt△ABC, ∠ ABC = 90, and BD is the height on the hypotenuse AC, then the projective theorem is as follows:

BD？ = Advertising CD

AB？ = exchange advertising

BC? =CD AC

It was put forward by Euclid, a famous mathematician in ancient Greece and the author of The Elements of Geometry.

In addition, the same is true when the triangle is not a right triangle, but the angle ABC is equal to the angle CDB. It can be proved by similarity, and the process is short.

Extended data:

Verification deduction

①CD？ = AD BD

②AC？ = AD AB

③ BC? = BD AB

④AC BC=AB CD

Proof: ①∫CD? +AD？ =AC？ ，CD？ +BD？ =BC？

∴2CD？ +AD？ +BD？ =AC？ +BC？

∴2CD？ =AB？ -Advertising? -BD？

∴2CD？ =(AD+BD)？ -Advertising? -BD？

∴2CD？ =AD？ +2AD BD+BD？ -Advertising? -BD？

∴2CD？ =2AD BD

∴CD？ =AD BD

②∵CD？ = ad BD (certification)

∴CD？ +AD？ =AD BD+AD？

∴AC？ =AD (BD+AD)

∴AC？ =AD AB

③ BC? =CD？ +BD？

BC? =AD BD+BD？

BC? =(AD+BD) BD

BC? =AB BD

∴BC？ =AB BD

④∫S△ACB =

AC×BC=

AB CD

∴

Ac BC=

AB CD

∴AC BC =AB CD