A Mathematical Blog

The coordinates of the midpoints of the sides of a triangle are (1, 2) (0, 1) and (2, 1), find its centroid. In any triangle, medians are the three segments connecting the vertices of the triangle to the midpoints of the sides opposite to the vertices. It turns out the three medians in any triangle always intersect at a single point. That point is called the centroid of the triangle.  The question is, can we find the coordinates of the centroid of a triangle from the given coordinates of the midpoints of the sides?  Yes, we can. The definition of the centroid doesn’t really give us any clue in that direction. But there are some properties of the centroid that can help us out here.   I don’t know of any formula or rule which directly relates the coordinates of the midpoints of the sides with the coordinates of the centroid. But I do know of a relation relating the coordinates of the centroid with the coordinates of the vertices of a triangle. The x and y...

Midpoint Theorem Midpoint theorem gives us the relation between the mid-segments and the edges of a triangle. A mid-segment of a triangle is the segment connecting the midpoints of any two sides.  In the figure above, MN is a mid-segment of △ABC. A triangle has three mid-segments in total.  The Midpoint theorem of triangles gives us a couple of relations. It says that the segment(mid-segment) joining the midpoints of any two sides of a triangle is  parallel to the third side of the triangle, and  half the length of the third side.  So in the above figure, by midpoint theorem, the length of mid-segment MN is half the length of side BC, and MN and BC are parallel line segments. So if BC is 10 units, MN is 5 units.  In mathematical terms, this can be written as :  In △ABC, MN is parallel to BC And MN = \(\frac{1}{2}\) BC …..(1) Let O be the midpoint of side BC in △ ABC.  MO and NO are the other two midsegments of △ ABC along with MN. And s...

Let’s take any triangle. We’ll call it the “primary triangle”. Join the midpoints of its three sides with line segments. The result will be a new triangle inside of the primary triangle. This new triangle obtained by connecting the midpoints of the sides of the primary triangle is called the midpoint triangle or medial triangle , not to be confused with the median triangle . If △PQR is the medial triangle of  △ ABC, then from the medial triangle’s point of view, the primary triangle △ ABC is said to be its anticomplementary triangle .  △PQR is the medial triangle of  △ ABC △ABC is the anticomplementary triangle of  △ PQR In an anticomplementary triangle the three midpoints of the sides are the three vertices of its medial triangle.  Can  △PQR have the medial triangle ? If M, N and O are the three midpoints of the sides of  △ PQR, then △ MNO will be the medial triangle of △ PQR. Or we could also say that △ PQR is the anticomplementary tri...

In this post, i will be talking about how to construct the incenter and the incircle of a triangle in Geogebra - an online graphing calculator.  Incenter is the center of the incircle of a triangle and is very easy to construct. It is also the point where the three angle bisectors of three interior angles of the triangle intersect. We could construct the incenter in Geogebra in the following simple steps(demonstrated with pictures) :  Read more about the Incenter here . Step 1 : Drawing a triangle. Select the tool Polygon under the Polygons section in the shapes menu.  In the graphing panel to the right, click on any three different spots for the three vertices of a triangle. Click again on the first point you made to complete forming the three sided triangle.  For constructing a triangle of three given side lengths or two given side lengths & one given angle, you can check out this post of mine. Step 2 : Select the tool Angle Bisector under the Constr...

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...