When I was given to write on the topics of Napoleon’s Theorem, the first thing that struck my mind was that it was somehow related to the French leader, Napoleon Bonaparte. But then a thought struck me: Napoleon was supposed to good at only politics and the art of warfare. Mathematics was never related to him. On surfing the internet to learn about the theorem, I came to know that this theorem was in fact named after the same Napoleon as he was good at Maths too (other than waging wars and killing people). The theorem was discovered in his ruling period and was named so in his honour.
Now moving on the towards the theorem part, it is a very observation made by some of the brightest minds of that time. Sorry to disappoint you, but the theorem…show more content… If we want to find height CM, it can be either aSinB or bSinA.
Thus Area = acSinB/2 or bcSinA/2 or abSinc/2.
Now coming back to Napoleon’s Theorem, consider the following figure where ABC is the original triangle and triangles FAB, EAC and CDH have been constructed on the sides of the triangle which have G, I and H as their respective centres. These centres are joined and a triangle GHI is obtained. Now all we have to do is to prove that the length of each side of triangle GHI is equal to s (which is considered to be the length GI).To help me in my proof, I gave taken length AG as t and length AI as u. Now since G, I and H are the respective centres of each equilateral triangle, they act as both incentres too. Thus the length t and u bisect the angles FAB and EAC into two 30 degree angles. Since we know the measure of angle GAB and IAC, we can now use the Cosine Law for the triangle AIG. It will be given as – s2 = t2 + u2 – 2tu(Cos(A+60))
On using formula Cos(A+B)= CosA CosB – SinA SinB, we get – s2 = t2 + u2 – 2tu(CosA Cos60 – SinA Sin60) s2 = t2 + u2 – tu(CosA) + tu√3(SinA)
Now we find t and u in terms of c and b.
From the above figure, we get to know…show more content… In triangular tiling, a tile of certain dimensions are taken and three other tiles representing equilateral triangles are taken and joined as stated in the Napoleon’s Theorem. Now the central triangle is fitted between any two of the equilateral triangles and the third equilateral triangle is fitted on the other side. Same procedure is repeated again and again until the whole floor is tiled. This tiling pattern is very pleasing to the eyes and was one of the most popular tiling patterns, where the central triangle is of a certain colour and the equilaterals are of shades of the central