If R and r is the Circumradius and Inradius of a non-degenerate triangle then due to Euler we have an Inequality stated as and the equality holds when the triangle is equilateral. This ubiquitous inequality occurs in the literature in many different equivalent forms  and also Many other different simple approaches for proving this inequality are known. (some of them can be found in , , , , and  ). In this article we present a proof for this Inequality based on two basic lemmas, one is on the fact “Among all the pedal’s triangles, the pedal triangle whose area is one fourth of area of the reference triangle has maximum area” and other is related to the “area of Ex-Touch triangle”.
Let ABC be a triangle. We denote its side-lengths by a, b, c, its semiperimeter by , its area by , its circumradius by , and its inradius by , and if D, E, F are the points of contact of excircles opposite to the vertices A, B, C( A-excircle, B-excircle and C-excircle) of the with the sides BC, AC, AB then BD = AE = s-c, CD = AF = s-b, BF = CE = s-a respectively.
1. Pedal’s Triangle:
More generally Pedal’s Triangle with respect to P is the triangle formed by joining the foot of the perpendiculars drawn from an arbitrary point P which lies in the plane of triangle to the sides.
2. Euler’s formula for area of pedal’s triangle:
If is the area of pedal’s triangle with respect to the point P and if…
Title: Application of Self-Organization Neural Network Technique (SOM) to Optimize Finite- Element Partial Differential Equation (PDEs) Results in Square-Shaped Structures Analysis.
The finite-element method (FEM) is a computationally method for solving partial differential equations (PDEs) with specific boundary conditions over a domain. When we applying the FEM to a domain, it has to divide to a finite number of elements and nodes. The collections of the elements and nodes form the…
An alternative modeling approach which remedies the computational aspect is differential equations. In contrast to the discrete event simulations, averaged quantities in case of large quantity production predict the time evolution of parts and include the dynamics inside the different production steps. To derive accurate continuous models the overall modeling goal is to transfer as much of the detailed and complex discrete model to the continuous level. This will be achieved regarding…
To give an another definition of complex number, we have to show z= ew for w, where z is any non-zero complex number. If we assume w = u + iv then
eueiv = |z|e^i arg z and this means that
|z| = e^u , v = arg z .
The equation |z| = e^u is a real equation, so we can write u = ln |z|, where ln |z| is
the ordinary logarithm with positive real numbers. Hence,
w = u + iv = ln |z| + i arg z = ln |z| + i(Arg z + 2n) , n = 0 , ±1 , ±2 , ±3 , . . .6
We profoundly examined the connection between…
1. What are the least, and most, amount of distinct zeroes of a 7th degree polynomial, given that at least one root is a complex number?
If the equation is 7th degree then it has 7 roots.
Those roots can be complex or real.
Complex roots always come in pairs, so if it has one, then it has 2, the other one being the conjugate of the first one. This in other words, if one complex root is a + bi, then the other complex root is a – bi.
If at least one root were complex, then we would have a…
Substituting , and into (31), and calculating the derivatives with respect to , we obtain
Since , we have
Hence there is a Hopf bifurcation at . We have the following result:
Theorem 13. Suppose holds. Then the system (29) undergoes a Hopf bifurcation when passes through emanating from the steady state leading to periodic solutions trajectories for either (super-critical bifurcation), or (sub-critical-bifurcation) or at .
investors will not pay for risk premiums, and defined by following stochastic differential equations:
Where: S(t) represents the price process of an underlying assets. V(t) is the variance of the corresponding instantaneous returns. And the initial condition S(0), V(0) should be strictly positive. r is the risk-free return rate (non-negative constant). k is the speed of the mean reversion. θ is the mean value of the variance. σ is the volatility of the variance. These three are positive…
We saw that the general term of our sequence was 1/n. So, as we keep increasing the value of n and eventually to infinity, the value of 1/n becomes zero. Thus, we fix that as our limit.
Therefore, 1/n<ɛ, implies that, 1/ɛ<n.
We know that n≥N.
Finally, we can claim that the sum of an infinite series is not just a distinct algebraic sum of the terms, but rather the “sum”, is a very special sum. It is obtained as the limit of the sequence of…
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus evaluate an antiderivative at the upper and lower limits of integration and take the difference. This theorem is separated into two parts. The first part is called the first fundamental theorem of calculus and states that one of the antiderivatives of some function may be obtained as the integral of the function with a variable bound of integration. The second part of the theorem, called the second fundamental theorem of…
MOOCs were a way for professors around the world to teach their classes online for free. Like most others, my thoughts on online education mostly revolved around second-rate scams and suspicious for-profit institutions (I’m looking at you, DeVry). However, curiosity got ahold of me, and I quickly signed up for several courses.
As it turned out, these courses were structured similarly to college classes; they consisted of weekly video lectures and a forum for collaboration between the…