0.1 Points and Circles of a Triangle
There are so many important points circles and circles in triangles. Knowing relationship between them can significantly help in solving many problems. Before we get into the first content, let’s clear up some symbols. To be honest, I just don’t think it would be efficient to define each points, so I will just mention them here. We will be using \(O\) for circumcenter, \(I\) for incenter, \(H\) for orthocenter, \(G\) for centroid, and \(I_A, I_B, I_C\) for excenters unless otherwise stated. With that in mind, let’s get started with Euler line and related theorem!
0.1.1 Servoir’s Theorem and Euler Line
As a warm-up, let’s start with Servoir’s Theorem and Euler Line. Fun fact, when you search Servoir’s Theorem online, you won’t probably get the theorem that we will discuss. I am not sure if we have an English name for this theorem or if it is even named, but this is actually very useful and Koreans call this Servoir’s Theorem.
Theorem 0.1.1: Servoir’s Theorem
For triangle \(ABC\) and midpoint \(M\) on side \(BC\), \(AH = 2OM\).
Proof.
First, consider the following diagram.
![[Picture]](/svg/MATH0103.260830x.svg)
From the diagram above, it is evident that lines \(\overline {AH}\) and \(\overline {DC}\) are parallel. Similarly, \(\overline {CH}\) and \(\overline {AD}\) are parallel. Therefore, quadrilateral \(AHCD\) is a parallelogram and \(AH = CD = 2OM\).
Continuing from the diagram above, we could also consider \(G\) of \(\triangle {ABC}\).
![[Picture]](/svg/MATH0103.260831x.svg)
Consider triangles \(AHG\) and \(MOG\). Notice that \(\angle (HAG) = \angle (OMG)\) since \(\overline {AH} \parallel \overline {OM}\). Moreover by Servoir’s Theorem, \(AH : OM = 2 : 1\) holds and \(AG : GM = 2 : 1\) is true by definition. Therefore, \(\triangle {AHG}\) and \(\triangle {MOG}\) are similar and \(\angle (AGH) = \angle (MGO)\). In other words, \(H, G, O\) are collinear, and we have a special name for the line!
Definition 0.1.2
The Euler Line of a triangle \(ABC\) is the line that contains \(H, G, O\).
As you can see, Euler line exists for all triangle except equilateral triangle. The fact that \(H, G, O\) is collinear will come very handy when solving problems!