*An easy but also deep journey on the geometry of a triangle and its exciting topics.
*

**[Under construction…]**

On this essay-course we begin studying some basic properties and gradually we touch some of the most interesting and “advanced” facts of the most simple plot in the Euclidean Geometry, the triangle. A skllied problem solver should be familiar with the biggest part of what we are going to mention. Nevertheless, we wish to show how simple the connection is between basic principles and advanced properties.

We did the same on the exercises in the end of this post. The first are very simple but we hope that the last will be tough even for skilled problem solvers.

All we need is an acute-angled triangle . The definitions of *middle points, medians, heights, circumcircles* are taken for granted. Thus we start from a very simple property:

**Lemma 1 – The most useful and fundamental lemma in plane geometry**

**The line segment joining the middle points of any two sides of a triangle is parallel to and equals half of the third side.**

*Proof*

We skip it. Don’t be afraid to prove this on your own, think as an entrance test for the rest. What you need is a parallelogram. Unfortunately, if you can’t prove this, you will meet troubles with what follows.

We move to our first theorem

**Theorem 1**

**The medians of a trianlge are concurrent; all pass through the same point.**

*Proof*

Let be the middle points of sides respectively. Suppose that segments meet at and that cuts in . We have to prove that . We take a point in the extension of segment so as . Applying Lemma 1 to triangles and we get that is parallel to and to . As a result, is a parallelogram and thus its diagonals, are bisected. The proof is now completed.

**Definition 1**

The point of intersection of the three medians of any triangle is called Centroid. From now on, we will represent this point with .

Hello again

This is something like re-introduction post. Lot of things have happened since my last posting. Participation in the International Mathematical Olympiad, Medals in many kinds of math olympiads, entrance in university, IMF in Greece, War in Syria. Of course the last two have nothing to do with this blog.

The important thing is that I am back. This means that from now on, this blog will be very active as regard to new posts and changes. So I hope it will also be active from your point of view, the viewers.

I would also like to mention my motivation for fast editing new posts. This blog had some unexpected visitors the past two months. When I noticed this in the stats, I was very suprised and happy. I must admit that I felt so happy as I was when I got informed for my bronze medal In the 52th IMO!

Anyway, I stop speaking and I start Geometry.

Enjoy!

Dear Reader

*(if you exist…)*

Recently, i succeded in the exams for the Greek team for Mathematical Olympiads. As a result i will participate (as GRE5) in the **28th Balkan Mathematical Olympiad **[4-8 May 2011] and in the **52nd International Mathematical Olympiad **[16-24 July 2011] (this is not sure but very possible). In the meanwhile, I will be giving school exams for university introduction. In addition, in August i “will have to take” some* *vacations!

So i will not be able to post something new before September. But i promise a dynamic comeback in Autumn, with many posts, difficult exercises, rich material for *Olympiad Geometry* and many PDF files with blog’s context.

Best regards,

*kalagz*

**About
**

Found by Ptolemy of Alexandria (or Claudius Ptolemaeus) (90-168 AD).

**Ptolemy’s First Theorem**

A quadrilateral ABCD is cyclic (inscribed) if and only if

or

“A quadrilateral is inscribed in a circle if and only if the sum of the products of its two pairs of opposite sides is equal to the product of its diagonals”

**Proofs **

**Proof 1 of the direct theorem**(Using a little Trigonometry :))

As in the figure,

since their sum is

We will use that and the formulas

,

(*Product to Sum formula),*

*(Sum to Product formula)*

*Cevian is called every segment that connects a vertice of a triangle with the opposite side. Three or more lines are concurrent when all pass through a common point, so we are talking about concurence.*

**Historical Facts**

This theorem was published from Giovanni Ceva (in honour of whom the segments before are named ) in 1678.

**Theorem**

In any triangle the cevians are concurrent if and only if

(simple form)

or

(trigonometric form)

**Proofs**

**Proof 1**

First we prove that we have concurrency if and only if . Assume that AD, BE, CF meet at point P. We extend BE and CF until they meet the parallel to BC from A at G and H respectively.

Then, we have the relations

(1) (2) and (3) Multipling relations 1, 2 and 3 we get the desired.

Now suppose that . We have to prove that AD, BE, CF are concurrent. Let BE and CF meet at P and suppose AP cuts BC at D’.

We get by what we have proved so far that . Dividing the last two relations we get that and since D and D’ both lie in segment BC, we get , QED

**Proof 2 **(Using areas)

We use the fact that the areas of triangles with equal altitudes are proportional to the bases of the triangles. Let AD, BE, CF concur at P. We have :

. Similarly we get and . Multipling these relations we get the desired .

For the converse we work like the first proof to get .

*Well, I start the collection with one of the most importand theorems in Geometry, the sines law for every triangle. I think this blog must start with this basic theorem because many others are proved usind this.
*

**Theorem**

In any triangle with angles and sides respectively the following is true

where is the circumradii of the triangle.

**theorems with proofs and applications in one site**. I’m just a school-boy who takes part in

**mathematical competitions**and loves geometry. My “addiction” to

**Classical Euclidean Geometry**made me create this blog. One more reason is to have an online collection with many theorems organized and “well-given” with some applications and exercises.

**historical facts**(if i know any) the theorem itself ofcourse,

**many proofs**,

**some remarks**, other

**results(findings)**and 3-4

**exercises**in increasing difficulty level taken from school books to Highly level Maths Olympiads.