(a) the natural proportion theorem:

(2) Proportional line segments in parallel lines:

(1) Proportional theorem of parallel lines divided into segments: the corresponding segments obtained by cutting two straight lines by three parallel lines are proportional (Figure 1, 2).

② A straight line parallel to one side of a triangle is directly proportional to the corresponding line segment obtained by cutting the other two sides (or extension lines of both sides) (Figures 3 and 4).

(3) a triangle cut by a straight line parallel to one side of the triangle and intersecting with the other two sides (or extension lines of both sides).

The three sides of the shape are proportional to the three sides of the original triangle (Figures 3 and 4).

(3) Proportional line segment in triangle:

① All corresponding line segments in similar triangles (corresponding edge, corresponding height, corresponding midline, corresponding bisector of angle, corresponding circumference).

The ratio of length …) is equal, which is equal to the similarity ratio.

(2) The ratio of all corresponding areas in similar triangles is equal, which is equal to the square of similarity ratio.

③ Pythagorean theorem: the square of the hypotenuse of a right triangle is equal to the sum of the squares of two right angles (Figure 5).

④ Projection theorem: The height on the hypotenuse of a right triangle is the proportional average of the projections of two right angles on the hypotenuse (Figure 5).

Any right-angled side of a right-angled triangle is the ratio of the median of its projection on the hypotenuse to the hypotenuse (Figure 5).

⑤ Sine theorem: In a triangle, the ratio of sine of each side to diagonal is equal (Figure 6). That is, /sinA=b/sinB=c/sinC.

⑥ Cosine Theorem: In a triangle, the square of any side is equal to the sum of the squares of the other two sides minus twice the cosine product of these two sides and their included angles (Figure 6).

For example, A2 = B2+C2-2B C Kossa.

(4) Proportional line segment in the circle:

Cyclic power theorem;

① Intersecting Chords Theorem The product of two intersecting chords divided by the intersection in a circle is equal (Figure 7).

(Inference: If the chord intersects the diameter vertically, half of the chord is the average of the ratio of the two line segments formed by its diameter. Figure 8)

② The tangent theorem leads to the tangent and secant of a circle from a point outside the circle, and the length of the tangent is the median term of the ratio of the lengths of two line segments from this point to the intersection of the secant and the circle (Figure 9).

(3) Secant theorem leads to two secant lines of a circle from a point outside the circle, and the length of each secant line is equal to the product of the intersection point of the circle (Figure 10).

(5) Operation of proportional line segment:

① Equal ratio or equal line segment replacement.

(2) Derived from the theorem of proportionality.

③ Algebraic or trigonometric methods are used for calculation.