Projective Theorem in Triangle

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 projections of this right angle on the hypotenuse and hypotenuse. ? The formula is as follows: As shown in the right figure, in Rt△abC, ∠ ACB = 90, and cd is the height on the hypotenuse AB, then the following projective theorem is obtained: ①CD? ; =AD DB，②BC？ =BD BA？ ,? ③AC？ =AD AB？ ; ? ④ AC BC = AB CD (equal product formula, available area proof)

AC*BC=2 S ABC

CD*AB=2 S ABC

AC*BC=AB*CD

abstract

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. In the formula Rt△abC, ∠ ACB = 90, and CD is the height on the hypotenuse AB, then the projective theorem is as follows: (1) (CD) 2; =AD DB，(2)(bc)^2； =BD BA，(3)(ac)^2； =AD AB. Equal product formula (4)ACXBC=ABXCD (proof of usable area)

Projective Theorem of Folding Right Triangle

The so-called projection is the projection of light. 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.

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:

Height theorem of right triangle

Folding proof

Solution:

In △BAD and △ACD,

∠∠Abd+∠Bad = 90, and ∠ CAD+∠ C = 90,

Schematic diagram of projective theorem

∴∠ABD=∠C，

∠∠BDA =∠BDC = 90。

∴△BAD∽△CBD

∴ AD/BD=BD/CD

Namely BD? = ad DC

The rest can be proved in the same way.

Height theorem of right triangle

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AB？ =AD AC，BC？ =CD CA

Two formulas have been added:

AB？ +BC？ =AD AC+CD AC =(AD+CD) AC=AC？ (namely Pythagorean theorem).

Note: AB? It means AB quadratic.

certificate

It is known that the middle angle of triangle A is 90 degrees and the AD is high.

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, and

BD=c cosB，CD=b cosC,∴a=BD+CD=b cosC+c cosB？ The rest can also be proved.

Prove 2: From 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。 ? The rest can also be proved.

Fold any triangle

The projective theorem of arbitrary triangle is also called "the first cosine theorem";

△ABC's three sides are A, B and C, and the angles they face are A, B and C respectively, so you have it.

a=b cosC+c cosB，

b=c cosA+a cosC，

c=a cosB+b cosA .

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.