What is the projective theorem and how to use it?

Projection is orthographic projection, and the vertical foot perpendicular to the bottom from a point to a vertex is called orthographic projection of that point on this straight line. The line segment between the orthogonal projections of two endpoints of a line segment on a straight line is called the orthogonal projection of this line segment on this straight line, that is, the projection theorem. 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. The formula is shown in the figure. At 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； Equal product formula (4) abxbc = bdxac proves that in △BAD and △BCD, ∠ A+∠ C = 90, ∠ DBC+∠ C = 90, ∴ A = ∠ DBC, ∞. = A.D. DC. Everything else is similar. Note: Pythagorean theorem can also be proved by the above projective theorem. From 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 any triangle is also called the "first cosine theorem": let the three sides of ⊿ABC be A, B and C, and the angles they subtend are A, B and C, then there are A = B COSC+C COSB, B = C COSA+A COSC, and C = A COSC. Note: Take "A = B COSC+C COSB" as an example. The projections of b and c on a are B COSC and C COSB, respectively, so there is a projective theorem. Proof 1: Let the projection of point A on the straight line BC be point D, then the projections of AB and AC on the straight line BC are BD and CD respectively, BD = CCCOSB, CD = b cosc, ∴ a = BD+CD = b cosc+c cosb. The rest can also be proved. Prove 2: From the sine theorem, we can get: b=asinB/sinA, c = asinc/sina = asin (a+b)/sina = a (sinacosb+cosasinb)/sina = acosb+(asinb/sina) cosa = a.cosb+b.cosa.