Projection theorem of mathematics in senior two.

Let's start with the definition of projection.

Projection: Orthographic projection, the vertical foot from a point to a straight line, is called the orthographic projection of the point on this straight line. The line segment between the orthographic projections of two endpoints of a line segment on a straight line is called the orthographic projection of the line segment on the straight line.

1. 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, for Rt△ABC, ∠BAC=90 degrees, and AD is the height on the hypotenuse BC. The projective theorem is as follows:

1.(ad) 2 = BD Special Zone,

2. (AB BC) 2 = BD,

3. (BC AC) 2 = CD

This is mainly introduced like a triangle, for example, (AD) 2 = BD DC:

Can be seen from the chart

△BAD is similar to△△△ ACD,

therefore

AD/BD=CD/AD，

So (ad) 2 = BD DC。

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

(AB) 2+(AC) 2 = (BC) 2, which is the conclusion of Pythagorean theorem.

Second, the arbitrary triangle projective theorem (also known as the "first cosine theorem"):

Let the three sides of the triangle abc be ABC, and the angles they face are ABC, then there are

a=b*cosC+c*cosB

b=c*cosA+a*cosC

c=b*cosA+a*cosB