A Mathematical Blog

I present a small catalog of images that I have designed in my time on this blog to support the textual content. I have designed most of the images on Adobe Draw/Adobe Fresnel application on my IPad, while few using the Geogebra software. There is no particular order of arrangement here; I thought it would be fun this way. :) Vector Components. 2 units in the positive x-direction(2i), 2 in the positive y-direction(2j), and 1 in the positive z-direction(1k).  u = 2i + 2j + k From post : Direction angles & Direction Cosines In two dimensional space, any given pair of lines can either be parallel or intersecting. However in three dimensions, a third case exists when a pair of lines can be neither of the two. Any non-intersecting, non-parallel pair of lines in 3D is a  pair of skew lines. From Post : Visualising the Shortest Distance between Skew Lines From Post : Medians and the Centroid of Triangle Orthic triangle is a triangle obtained by joining the foot of the three altitudes o

Another one of popular and important theorems in circles is the one which states that the angle in a semicircle is a right angle .                                   OR The angle subtended by a diameter at any point on the circumference of a circle is a right angle . Both are just two different ways of stating the same thing.  This theorem is actually a direct outcome of another theorem on circles which was also the topic of my previous post. So let’s quickly review the theorem first that I wrote about in my last post. To avoid confusing my brain, I am gonna call it Theorem 1. And the theorem whose statement is mentioned above as Theorem 2, which I will show you is a special case of Theorem 1.  Let’s quickly get familiar with some basic terms. An arc is any part of the circumference(boundary) of a circle. In the given figure, AFB is an arc of the circle with center O. It is a minor arc because it is smaller than half the length of the circumference of the circle.  Let P b

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 coordinates of a c