**what is the Euclid’s theorem?**

He **euclid’s theorem** demonstrates the properties of a right triangle by drawing a line that divides it into two new right triangles that are similar to each other and similar to the original triangle; So, there is a proportional relationship.

Euclid was one of the greatest mathematicians and geometers of the ancient age who made several proofs of important theorems. One of the main ones is the one that bears his name, which has had a wide application.

This has been so because, through this theorem, it explains in a simple way the existing geometric relationships in the right triangle, where the legs of this are related to their projections on the hypotenuse.

**Formulas and proof**

Euclid’s theorem proposes that in every right triangle, when a line is drawn —which represents the height that corresponds to the vertex of the right angle with respect to the hypotenuse— two right triangles are formed from the original one.

These triangles will be similar to each other and will also be similar to the original triangle, which means that their similar sides are proportional to each other:

The angles of the three triangles are congruent; that is to say, when they are rotated 180 degrees on their vertex, one angle coincides on the other. This implies that they will all be equal.

In this way, the similarity that exists between the three triangles can also be verified, due to the equality of their angles. From the similarity of triangles, Euclid establishes the proportions of these from two theorems:

height theorem.

The legs theorem.

This theorem has a wide application. In Antiquity it was used to calculate heights or distances, representing a great advance for trigonometry.

It is currently applied in various areas that are based on mathematics, such as engineering, physics, chemistry and astronomy, among many other areas.

**height theorem**

In this theorem it is established that in any right triangle, the height drawn from the right angle with respect to the hypotenuse is the geometric proportional mean (the square of the height) between the projections of the legs that it determines on the hypotenuse.

That is, the square of the height will be equal to the multiplication of the projected legs that form the hypotenuse:

hc2 = m*n

**Demonstration**

Given a triangle ABC, which is right at vertex C, plotting the height produces two similar right triangles, ADC and BCD; therefore, their corresponding sides are proportional:

In such a way that the height hc that corresponds to the segment CD, corresponds to the hypotenuse AB = c, thus it is had that:

In turn, this corresponds to:

Clearing the hypotenuse (hc), to multiply the two members of the equality, it is necessary to:

hc * hc = m * n

hc2 = m*n

Thus, the value of the hypotenuse is given by:

**Legs theorem**

This theorem establishes that, in any right triangle, the measure of each leg will be the geometric proportional mean (the square of each leg) between the measure of the hypotenuse (complete) and the projection of each one on it:

b2 = c*m

a2 = c*n

**Demonstration**

Given a triangle ABC, which is right at vertex C, in such a way that its hypotenuse is c, when drawing the height (h) the projections of the legs a and b are determined, which are the segments m and n respectively, and which lie on the hypotenuse.

Thus, it is found that the height drawn on the right triangle ABC generates two similar right triangles, ADC and BCD, so that the corresponding sides are proportional, like this:

DB = n, which is the projection of the leg CB onto the hypotenuse.

AD = m, which is the projection of the leg AC onto the hypotenuse.

Then, the hypotenuse c is determined by the sum of the legs of its projections:

c = m + n

Due to the similarity of the triangles ADC and BCD, we have that:

The above is the same as:

Clearing the leg «a» to multiply the two members of the equality, it is necessary to:

a*a=c*n

a2 = c*n

Thus, the value of leg «a» is given by:

In the same way, due to the similarity of the triangles ACB and ADC, we have that:

The above is equal to:

Clearing the leg «b» to multiply the two members of the equality, it is necessary to:

b*b=c*m

b2 = c*m

Thus, the value of leg «b» is given by:

**Relationship between Euclid’s theorems**

The theorems with reference to height and the legs are related to each other because the measure of both is made with respect to the hypotenuse of the right triangle.

Through the relationship of Euclid’s theorems the value of the height can also be found; that is possible isolating the values of m and n from the theorem of the legs and they are replaced in the theorem of the height. In this way it is fulfilled that the height is equal to the multiplication of the legs, divided by the hypotenuse:

b2 = c*m

m = b2 ÷ c** **

a2 = c*n

n =a2 ÷ c

In the height theorem replace myn:

hc2 = m*n

hc2 = (b2 ÷ c) * (a2 ÷ c)

hc = (b2 * a2) ÷ c

**solved exercises**

**Example 1**

Given the triangle ABC, right-angled at A, determine the measure of AC and AD, if AB = 30 cm and BD = 18 cm

**Solution**

In this case, we have the measurements of one of the projected legs (BD) and of one of the legs of the original triangle (AB). In this way, the leg theorem can be applied to find the value of leg BC.

AB2 = BD * BC

(30)2 = 18 * BC

900 = 18 * BC

BC = 900 *÷ *18

BC = 50cm

The value of the leg CD can be found knowing that BC = 50:

CD = BC – BD

CD = 50 – 18 = 32 cm

Now it is possible to determine the value of leg AC, once again applying the leg theorem:

AC2 = CD * BD

AC2=32*50

AC2 = 160

AC = √1600 = 40cm

To determine the value of the height (AD) the height theorem is applied, since the values of the projected legs CD and BD are known:

AD2=32*18

AD2 = 576

AD = √576

AD = 24cm

**Example 2**

Determine the value of the height (h) of a triangle MNL, rectangle in N, knowing the measurements of the segments:

NL = 10cm

MN = 5cm

PM = 2cm

**Solution**

You have the measure of one of the legs projected onto the hypotenuse (PM), as well as the measures of the legs of the original triangle. In this way, the Legs Theorem can be applied to find the value of the other projected leg (LN):

NL2 = PM * LM

(10)2 = 5 * LM

100 = 5 * LM

LP = 100 *÷ *5 = 20

Since the value of the legs and the hypotenuse is already known, through the relationship of the theorems of the height and the legs, the value of the height can be determined:

NL = 10

MN = 5

LM = 20

h = (b2 * a2) ÷ c.

h = (102 * 52) *÷ *(twenty)

h = (100 * 25) *÷ *(twenty)

h = 2500 *÷ *twenty

h = 125 cm.