An acute-angled triangle ABC is given. Let AD, CE, BM be its heights, CD = DE = 7, DM = 8. Find CB.

Take an acute-angled triangle ABC with heights AD, BM and CE. Let’s connect point D with the bases of other heights M and E. By the problem statement:

| CD | = | DE | = 7;

| DM | = 8;

It is necessary to calculate the length of the aircraft side. Knowing that | CD | = 7, the problem is essentially reduced to calculating the length of the segment BD.

Similarity properties of triangles
Two triangles are considered similar if

the ratio of the lengths of their respective two sides is the same and the angles between them are equal;
the inner angles of one triangle are equal to the corners of the second triangle;
the ratio of the lengths of all three sides of one triangle to the lengths of the sides of the second triangle is the same.
In our case, consider two triangles BED and CDM. According to the conditions of the problem:

| EB | = | Sun | cos (B), | BD | = | AB | cos (B);

and, therefore, | EB | / | ВD | = | Sun | / | AB |. This means that triangles BED and ABC are similar.

Further, for triangle CDM we have:

| CD | = | AC | cos (C), | MC | = | BC | cos (C);

and therefore | CD | / | MC | = | AC | / | ВC |. This means that triangles CDM and ABC are similar.

Calculation of the length of the BC side
Since both triangles BED and CDM are similar to triangle ABC, these triangles are similar to each other. It follows from this similarity that:

| DE | / | ВD | = | CD | / | DM |

Substituting the initial data, we get:

7 / | ВD | = 7/8 or | ВD | = 8.

As a result, we get:

| BC | = | CD | + | BD | = 7 + 8 = 15

Answer: the length of the BC side is 15.