EUCLID'S THEOREM: PROOF, APPLICATION AND EXERCISES - MATHS - 2025

The Euclid's theorem shows the properties of a triangle to the draw a line that divides it into two new triangles that are similar and, in turn, are similar to the original triangle; then, there is a relationship of proportionality.

Euclid was one of the greatest mathematicians and geometricians of the ancient age who performed 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 the case because, through this theorem, it explains in a simple way the geometric relationships existing in the right triangle, where the legs of the triangle are related to their projections on the hypotenuse.

Formulas and demonstration

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.

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 about their vertex, one angle coincides with the other. This implies that they will all be the same.

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

- Height theorem.

- Theorem of the legs.

This theorem has a wide application. In ancient times 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:

h _c ² = m _* n

Demonstration

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

In such a way that the height h _c that corresponds to the segment CD, corresponds to the hypotenuse AB = c, thus we have:

In turn, this corresponds to:

Solving for the hypotenuse (h _c), to multiply the two members of the equality, we have:

h _{c *} h _{c =} m _* n

h _c ² = m _* n

Thus, the value of the hypotenuse is given by:

Leg theorem

In this theorem, it is established that, in every 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:

b ² = c _* m

a ² = c _* n

Demonstration

Given a triangle ABC, which is right at the vertex C, in such a way that its hypotenuse is c, when plotting 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, the height plotted 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 leg CB onto the hypotenuse.

AD = m, which is the projection of the leg AC on 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:

The above is the same as:

Solving for leg “a” to multiply the two members of the equality, we have:

a _* a = c _* n

a ² = 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:

The above is equal to:

Solving for leg "b" to multiply the two members of the equality, we have:

b _* b = c _* m

b ² = c _* m

Thus, the value of leg "b" is given by:

Relationship between Euclid's theorems

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

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

b ² = c _* m

m = b ² ÷ c

a ² = c _* n

n = a ² ÷ c

In the height theorem we replace m and n:

h _c ² = m _* n

h _c ² = (b ² ÷ c) _* (a ² ÷ c)

h _c = (b ² _* a ²) ÷ c

Solved exercises

Example 1

Given the triangle ABC, right 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.

AB ² = BD _* BC

(30) ² = 18 _* BC

900 = 18 _* BC

BC = 900 ÷ 18

BC = 50 cm

The value of 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, applying the leg theorem again:

AC ² = CD _* BD

AC ² = 32 _* 50

AC ² = 160

AC = √1600 = 40 cm

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:

AD ² = 32 _* 18

AD ² = 576

AD = √576

AD = 24 cm

Example 2

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

NL = 10 cm

MN = 5 cm

PM = 2 cm

Solution

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

NL ² = PM _* LM

(10) ² = 5 _* LM

100 = 5 _* LM

PL = 100 ÷ 5 = 20

As 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 = (b ² _* a ²) ÷ c.

h = (10 ² _* 5 ²) ÷ (20)

h = (100 _* 25) ÷ (20)

h = 2500 ÷ 20

h = 125 cm.

References

1. Braun, E. (2011). Chaos, fractals and weird things. Fund of Economic Culture.
2. Cabrera, VM (1974). Modern Mathematics, Volume 3.
3. Daniel Hernandez, DP (2014). 3rd year math. Caracas: Santillana.
4. Encyclopaedia Britannica, i. (nineteen ninety five). Hispanic Encyclopedia: Macropedia. Encyclopedia Britannica Publishers.
5. Euclid, RP (1886). Euclid's Elements of Geometry.
6. Guardeño, AJ (2000). The legacy of mathematics: from Euclid to Newton, the geniuses through their books. Sevilla University.