FIRST HALF

The **league match** between **Real Madrid** and **F.C. Barcelona** will take place in a few days.

vs. |

**José Mourinho** wants to **plug** the **gaps** in **midfield**, and let less space for **Barcelona** midfielders.

To achieve this, he wants to position his players as follows:

**Xabi Alonso** will play in the centre of the pitch. **Di Maria** will play down **the right**, **40 m.** **ahead** of **Xabi Alonso**, and **30 m.** from the **centre of the field**. Also **40 m.** **ahead** of **Alonso**, but on the **left wing**, and also **30 m.** away from the **centre of the field**, we'll find **Cristiano Ronaldo**.

And **Mesut Özil** will play in an **intermediate position** between **Xabi Alonso** and the line **Di María** - **Cristiano**, and also **in the centre** of the pitch.

**Mourinho** wants to **minimize** the **distance** between **Özil** and the other **3 players**, so they can do a better **pressure** on the ball out of **Barcelona** and better cover the gaps.

Thus, the **exact position** of **Özil** will be that in which the **sum** of the **distances** between **Özil** and the other **3 players** is the **minimum possible**.

Can you **figure out** in which **exact point** **Özil** should be placed?

Some people would have thought that **Özil** should be placed on the **line** **Cristiano Ronaldo** - **Di María**. Others would reply that he should be placed on the **centroid** (intersection of the medians or center of gravity) of the **isosceles triangle** formed by the other **3 players**. Others will think that he should be at the **circumcenter** (center of the circumcircle, which passes through the three vertices of the triangle), on the **incenter** (center of incircle, tangent to the sides of the triangle), or even on the **orthocenter** (intersection point of altitudes).

But **any** of these solutions is **correct**. The result must be found another way. Let's see how:

We have **Alonso** (A), **Cristiano** (C) and **Di María** (D) situated in the pitch as follows:

We put now **Özil** (O) at any **point** of the **altitude of the triangle**, since according to the instructions of **Mourinho** he should be ahead of **Alonso** and the center of the pitch:

Our goal is to **minimize** **the sum of the distances to the other 3 players**, that is, achieving a **minimal result** for the **sum** of the segments ÖA + ÖC + ÖD.

To do this, we **turn the triangle** **ACD** **60º** counterclockwise on **Di María**:

We now look at these 2 **triangles**: A'Ö'D' and ÖAD. We can verify that ÖA measures the same as A'Ö', since both triangles are **identical**, except by a **60º** rotation.

Now we will focus on the **triangle** DÖ'Ö. We know that the angle **α** equals 60º, and that DÖ and DÖ' have the **same measure**. Therefore, DÖ'Ö necessarily has to be an **equilateral triangle**, from which it's derived that DÖ and Ö'Ö have the same length.

With all this data, we have already **solved** the problem. We know that OA = O'A' and that DO = OO'. Therefore, the **sum of the distances** (S) between **Özil** and his 3 playmates is equal to: **ÖA** + **ÖD** + **ÖC** = **A'Ö'** + **Ö'Ö** + **ÖC**.

Let's see what this **sum** corresponds to in our **graph**:

Indeed, this is the length of the **way** from A' to C.

And what we want is to reduce this **distance** to the **minimum** possible. To do this, we use the well-known axiom which states that 'the **shortest distance** between 2 points is a **straight line**'.

So if we want to **minimize** the **distance** between A' and C, we must position **Özil** in the **line** connecting the 2 points. Thus, when we turn the triangle 60º, the **paths** šA'Ö' + Ö'Ö + ÖC and **A'C** overlap.

And as we know that **Mourinho** wants to put **Özil** at the center of the field, on the **altitude** of the **triangle ADC**, we have to position him on the point where this **altitude** intersects the line **A'C**.

In case that **Özil** could move to any point within the triangle formed by the other 3 players, but not necessarily in the center of the field, we should apply the same procedure. This way, we know that **Özil** should be placed on the line **A'C**, and to determine the exact point in it, we would proceed analogously: we **turn** the figure **60º**, this time on **Cristiano Ronaldo** and clockwise. And so we get a new **line** between **D** and **A''**, which minimizes the path between the 2 extremes. And the **point** where the 2 **lines** **intersect** is where we have to place **Özil**.

Thus we have found the **exact point** where **Mourinho** wants to place **Özil** in the pitch, in order to counter the game of **FC Barcelona** midfielders.

Let's see it now from **another quantitative point of view**.

We've got an **isosceles triangle** ADC, and its **altitude** **AB**. We know that **Özil** is to be located in the **center** of the pitch, ahead of **Xabi Alonso**, that is, in a point of the **segment** AB.

If we call x the **distance** from **Özil** to **point B**, **Özil** is at a distance of **40-x** from **Xabi Alonso**.

Our mission is to find our at which **point** the distance **ÖA** + **ÖD** + **ÖC** is **minimal**.

We know that **ÖA** is equal to **(40-x)**, and that **BD** equals **30**.

By applying Pythagorean theorem, we have:

And we also know that **ÖC** = **ÖD**

With these data we set a **function** f(x) which **calculates** the **sum** of the **3 segments** based on where we put **Özil**

We want to find **x** such that **f(x)** is **minimal**. This will happen when the **derivative** function of **x** is equal to **0**, that is, when **f'(x) = 0**.

So we get

We could check that this **point** corresponds to the point from which you can **see** the **3 sides** of the triangle from a **120º** **angle**.

And likewise we can also check that the **situation** of this **point** **doesn't depend** on the distance from **Xabi Alonso**. The point will be the same if he is **40 m.** far from the line between **Cristiano** and **Di María** that if he's only **20 m.** far from it. It only **depends on the separation** between these **2 players** (CD = 2 BD), because:

The point where **Özil** should be placed is called **the Fermat point** (also called **Torricelli point**). It`s the first **notable point** of the triangle that was found after the time of **Euclid**.

Named in honor of the 17th century **French mathematician** **Pierre Fermat** who posed the following problem to the **Italian scientist** **Evangelista Torricelli** (1608 - 1647): 'Given an **acute-angled triangle ABC**, construct a **point P** such that the **sum of the distances** from it **to the 3 vertices** A, B and C is **the minimum possible**'.

Some people say that it was not **Torricelli** who solved the problem, but a **disciple** called **Vicenzo Viviani**, who published the solution in his name, in 1659. And others attribute the approach and its solution to **Jakob Steiner** (1796-1863). In addition, there have been many mathematicians who have been studying this problem, as **Hofmann**, with his graphic demonstration in 1929, or **Alfred Weber**,
who in 1909 studied the calculus of the **optimal location** from an economic point of view so to minimize the weighted sum of the distances from a place to a set of given points, or **Simpson** (1710-1761), with his geometric proof, or **Varignon** and his machine.

The first solution we propose in this case corresponds to that published by **Joseph Ehrenfried Hofmann** in 1929, which was also discovered independently by **Tibor Gallai** among others.

The second solution is only valid for **isosceles triangles**, and assuming that **Özil** is in the **altitude** of the triangle.

In any case, this seemingly inconsequential **Fermat point** assumes an especial **importance** in various fields of **science and technics**. So, when we want to build a **road** connecting 3 or more cities, we will apply this theorem to find the ideal line. Or when a company with 3 production centers wants to establish a headquarters in an optimal location to **minimize** **transport costs**. Or if we want to find the centroid or center of mass of a set of masses. Or for the optimal design of **electrical circuits** and **telecomunication networks**...

And if we **increase the number of points** between which we want to **optimize their connection**, we find the **Steiner tree problem**, which is a combinatorial optimization problem for seeking the shortest interconnection for a given set of elements. The **Fermat point** also gives a solution to the general problem of the **geometric median**, when the number of points in the plane is equal to 2, and to the Steiner tree problem for 3 points.

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