Mathematical photography theorem model

The height theorem of right triangle means that in a right triangle, the height on the hypotenuse is the middle term of the ratio of the projection of two right angles on the hypotenuse, and the right angle is the middle term of the ratio of the projection of this right angle on the hypotenuse to the hypotenuse. The projection theorem of right triangle, also known as Euclid theorem, is that in a right triangle, the height on the hypotenuse is the proportional average of the projections of two right angles on the hypotenuse, and each right angle is the proportional average of the projection of this right angle on the hypotenuse and the hypotenuse.

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)？ = ad DC, ②(AB)? =AD AC，(3)(BC)？ =CD CA .

Equal product formula (4)AB×BC=AC×BD (can be proved by "area method" or similar method) (5)(AB)? /(BC)? = Advertisement/CD

Proof of projective theorem of right triangle: (mainly calculated from similarity ratio of triangle)

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

Schematic diagram of projective theorem (geometric sketchpad) ∴∠ABD=∠C,

∠∠BDA =∠BDC = 90。

∴△BAD∽△CBD

∴ AD/BD=BD/CD

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

The projective theorem is as follows:

AB？ =AD AC，BC？ =CD CA

Two formulas have been added:

AB？ +BC？ =(AD AC)+(CD AC) =(AD+CD) AC=AC？ .

Pythagoras proof projection

∵AD？ =AB？ -BD？ =AC？ -CD？ ,

∴2AD？ =AB？ +AC？ -BD？ -CD？ =BC？ -BD？ -CD？ =(BC+BD)(BC-BD)-CD？ =(BC+BD)CD-CD？ =(BC+BD-CD)CD=2BD×CD。

So AD? =BD×CD。

Using this conclusion, we can get: AB? =BD？ +AD？ =BD？ +BD×CD=BD×(BD+CD) =BD×BC，

AC？ =CD？ +AD？ =CD？ +BD×CD=CD(BD+CD)=CD×CB。

To sum up, the projective theorem is obtained. It can also be proved by the knowledge of triangle area.