Problems

It is often the case in geometric situations that figures look very similar, but not quite equal. Two polygons on a plane are called similar, if and only if ALL their corresponding angles are equal AND the ratio between ALL the corresponding sides is the same.
The relation between the corresponding sides, in our case it is \(\frac{AB}{IH}\) is called the similarity coefficient between the figures. It is common practice to write vertices of similar figures in the order that respects the similarity.

We will derive some criterions for determining similarity and apply them to solve problems. This means the statement of some examples and problems may be helpful for problems coming after them!

Let \(ABC\) and \(DEF\) be such triangles that angles \(\angle ABC = \angle DEF\), \(\angle ACB = \angle DFE\). Prove that the triangles \(ABC\) and \(DEF\) are similar.

The medians \(AD\) and \(BE\) of the triangle \(ABC\) intersect at the point \(F\). Prove that the triangles \(AFB\) and \(DFE\) are similar. What is their similarity coefficient?

In a triangle \(\triangle ABC\), the angle \(\angle B = 90^{\circ}\) . The altitude from point \(B\) intersects \(AC\) at \(D\). We know the lengths \(AD = 9\) and \(CD = 25\). What is the length \(BD\)?

Let \(ABC\) and \(DEF\) be two triangles such that \(\angle ACB = \angle DFE\) and \(\frac{DF}{AC} = \frac{EF}{BC}\). Prove that triangles \(ABC\) and \(DEF\) are similar.

Let \(AA_1\) and \(BB_1\) be the medians of the triangle \(ABC\). Prove that triangles \(A_1B_1C\) and \(BAC\) are similar. What is the similarity coefficient?

Let \(AD\) and \(BE\) be the heights of the triangle \(ABC\), which intersect at the point \(F\). Prove that the triangles \(AFE\) and \(BFD\) are similar.

Let \(AD\) and \(BE\) be the heights of the triangle \(ABC\). Prove that triangles \(DEC\) and \(ABC\) are similar.

The area of the red triangle is \(25\) and the area of the orange triangle is \(49\). What is the area of the trapezium \(ABCD\)?

Prove that the ratio of perimeters of similar polygons is equal to the similarity coefficient.