The content of projective theorem

The Projective Theorem of Right Triangle

The formula is shown in the figure. At Rt△ABC, ∠ ABC = 90, and BD is the height on the hypotenuse AC, then there is a projective theorem as follows:

( 1)(bd)^2； = ad DC,

(2)(ab)^2； =AD AC，

(3)(bc)^2； =CD AC .

It is proved that in △BAD and △BCD, ∠ A+∠ C = 90, ∠ DBC+∠ C = 90, ∴∠A=∠DBC and ∠ BDA. = A.D. DC. Everything else is similar. (It can also be proved by Pythagorean theorem)

Note: Pythagorean theorem can also be proved by the above projective theorem. According to formula (2)+(3):

(ab)^2； +(bc)^2； = ad AC+CD AC =(ad+CD)ac=(ac)^2； ,

That is (ab) 2; +(bc)^2； =(ac)^2； .

This is the conclusion of Pythagorean theorem. [Edit this paragraph] The projective theorem of arbitrary triangle.