What is the projective law and how to prove it; what should I do if I don’t understand the question about the equal product formula?

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Projection Theorem

Theorem: In a right triangle, the height on the hypotenuse is the median of the ratio of the two right-angled sides projecting onto the hypotenuse, A right-angled side is the median of the ratio between the projection of this right-angled side on the hypotenuse and the hypotenuse.

Among them, the vertical foot of a perpendicular line drawn from a point to a straight line is called the orthographic projection of this point on this straight line. A 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. Projection theorem;

Suppose right triangle ABC, AB is the hypotenuse and CD is the height, then

The square of AC = AD×AB

The square of CB = BD×BA

The square of CD = AD×DB

Equal product formula;

AD×AC=AB×AC

Explanation ;AC/AB=AC/AD (proportional formula)

∠ACB=90°, CD⊥AB, then AC^2=AD×AB, BC^2=BD×AB,CD^2= AD×BD. The above proportional formulas are collectively called the projection theorem. It is mainly used to solve the problem of the hypotenuse of a right triangle and the connection between the fixed point and the hypotenuse. For example, given the lengths of AD and BD, find AC:BC.

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