I. Introduction to similar triangles
Two triangles with three sides in proportion are called similar triangles. Similar triangles is one of the important proof models in geometry, and it is a generalization of congruent triangles. Congruent triangles can be understood as similar triangles with a similarity ratio of 1. Similar triangles is actually a set of theorems, which mainly describes the relationship between edges and angles in two triangles in similar triangles geometry.
Second, the nature of similar triangles
The corresponding angles of similar triangles are equal, and the corresponding sides are proportional. The ratio of all corresponding line segments (corresponding height, corresponding midline, corresponding angle bisector, circumscribed circle radius, inscribed circle radius, etc.). ) is equal to similarity ratio in similar triangles. The ratio of similar triangles perimeter is equal to the similarity ratio. The ratio of similar triangles area is equal to the square of similarity ratio.
Similar triangles inscribed circle, circumscribed circle diameter ratio and perimeter ratio are all similarity ratios, and the area ratio of inscribed circle and circumscribed circle is the square of similarity ratio. Similarity ratio is equal to the arithmetic square root of area ratio. Not necessarily in the same plane triangle.
Introduction of similar triangles's reasoning, parallel cutting theorem and projective theorem;
1, similar triangles inference
The waist and bottom are similar to two isosceles triangles. Two right triangles divided by the height on the hypotenuse are similar to the original triangle. If the two sides of a triangle and the median line of either side of the triangle are proportional to the corresponding parts of another triangle, then the two triangles are similar.
2. Parallel cutting theorem
Two straight lines intersect a set of parallel lines, which are proportional to the corresponding line segments of this set of parallel wire cuts. A straight line parallel to one side of a triangle cuts the other two sides, and the cut triangle is proportional to the corresponding side of the original triangle. If a set of parallel lines cut on a straight line are equal, then the line segments cut on any straight line (intersecting with this set of parallel lines) are also equal.
3. Projective theorem
The projective theorem (also known as Euclid theorem) is commonly known as the mother-child triangle: in a right triangle, the height on the hypotenuse is the middle term of the ratio of the projection 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.