The inscribed triangle and the largest square is a common problem in real life. Algebraic method is a common method to solve related problems. Algebraic geometry never separates. It is also an important mathematical idea to explore the geometry of problems by using what you have learned.

Similar triangles's principle is the most commonly used algebraic method to inscribed the largest square in a triangle, which is also the most easily thought of. However, this method involves measuring and calculating the error caused by a ruler, so the accuracy of the square finally made is discounted, and the ruler drawing can greatly reduce the error and improve the accuracy.

First of all, let's recall a conceptual similarity graph: if two graphs are not only similar graphs, but also the connecting lines of each group of corresponding edges intersect at a point, then these two graphs are called similarity graphs, and this point is called similarity center. For example, the following two graphs are similarity graphs, and the similarity center is 0.

As can be seen from the concept and image, a position figure includes three elements: two figures and a center. Obviously, if you want to make one of the figures, you only need to know the positions and similar centers of the two figures. Let's cut to the chase.

Make the largest inscribed square of △ABC:

Using the principle of similarity graph, we can make a triangle inscribed with the largest square by choosing a similar center and then making that square. How do you find this similar center and this square?

The center of exploring similarity:

The center of the position is a point. Where is the better point? Look at this picture-a triangle with three vertices. It is most natural to choose one of the vertices as the center of similarity, so choose point A. ..

Explore another square: where is it? Too abstract! ? Don't! ! We should also start with the concept of similar graphics.

The similarity center is a line connecting the corresponding vertices, that is, there are two vertices on each line. We know that the largest square inscribed in a triangle has two vertices on AB and AC respectively, and A is the similar center, so the vertex of the other square must also be on AB, AC or the extension line of AB and AC. We don't know the exact location of the largest square inscribed in the triangle for the time being, but we can guess its approximate location to help us explore.