What is the projective theorem? How to understand?

Simply put? It can be proved by triangle similarity.

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

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？ . [ 1]

Prove projective theorem with Pythagorean theorem

∵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。