A Mathematical Blog

In the plane of any given triangle there are nine points that are always concyclic; a circle can be drawn passing through all of them. This circle is called the nine-point circle of that triangle.  These nine points are  : Feet of the altitudes Midpoints of the sides Midpoints of the three segments from vertices to the orthocenter The center of the nine-point circle is called the nine-point center. Here’s a step by step guide on how to construct the nine-point circle for a triangle in  Geogebra - an online Graphing Calculator(demonstrated with pictures).  We are gonna need a triangle to begin with. I will assume you already know how to construct triangles in Geogebra. But just in case if you don’t, check out this post of mine where you will find ways to construct triangles of three given side lengths or two given side lengths & one given angle. The easiest way to construct any casual triangle is by using the Polygon tool. Step 1 : In the editing panel o...

In this post, we are going to see how to construct the Euler line in Geogebra - an online graphing calculator. But first, a short description of what the Euler line is.  In any triangle, the four notable centers - centroid, orthocenter, circumcenter and nine-point center - are collinear; meaning a straight line can connect all four of them.  This line is the Euler line of that triangle.  To see that by ourselves, we are going to use 2D graphing calculator in Geogebra to construct these four centers. So here’s a step by step guide on how to draw the Euler line passing through these four centers of a triangle in Geogebra (demonstrated with pictures).  We are gonna need a triangle to begin with. I will assume you already know how to construct triangles in Geogebra. But if you want to know how to construct triangles of three given side lengths or two given side lengths & one given angle in Geogebra, check out this post of mine.  Step 1 : Drawing a ...

The Euler Line of a triangle is simply a straight line that passes through four of the commonly known centers of the triangle. They are,  Circumcenter(C) Centroid(G) Orthocenter(H) Nine-point center(L) It creates more of an impact if we say it the other way around. That is, for any given triangle the above mentioned four centers are always collinear; they lie on the same line - the Euler Line.  Things are slightly different for an equilateral triangle. These four centers are the same single point. We can draw infinitely many lines passing through a single point. And so it only makes sense that the concept of 'the' Euler line is associated only with non-equilateral triangles.  Now for those of you reading this who don't know about these four centers of a triangle, let me quickly give a short description of each one of them.  Circumcenter is the center of the circumcircle of a triangle. It is the circle through the three vertices of the triangle. Circumcen...