Ask a master to answer math questions

This question cured my cervical spondylosis for many years [laughs]. . . .

Law 1: Rogue special circumstances law.

Obviously E and F are moving points and the ratio is a constant value. So take a special case to study.

Let E and B coincide, then F and A coincide. The ratio to be found is BD/AC.

It should go without saying below. BD=10 (Pythagorean theorem) AC=9.6 (triangle equal area method)

So DE/CF=BD/AC=25/24

Method 2: Haven’t thought about it yet How to name it.

Obviously, the four points A, B, C, and D form a circle, so the system is established as shown in the figure.

The equation of the circle is x? y?=25, and it is easy to find A, The coordinates of point C. So we get the equations of straight lines AB and AD.

Suppose the coordinates of points E and F are (x1, y1) (x2, y2) respectively (x1 and x2 have ranges)

Then the coordinates of points E and F satisfy AB and AD Equation (①②), and the slope product of DE and CF is -1 (③)

Four unknowns and three equations should be able to get the relationship between the unknowns (that is, they are all represented by one unknown number) )

The ratio to be found is represented by two-point coordinates, and then replaced by an unknown number. If the data for this question is gathered together, the ratio can be found to be 25:24.

(I am too lazy, I only post methods)