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**, whenever the angle that Xabi draws with the other 2 players is smaller than 120º (in such case, Özil should stand at the same place in the pitch than Xabi Alonso, in the vertex). The point will be the same if Xabi 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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