Point of Feuerbach

Posted July 5th, 2008 by Structure

Let ABC a triangle and consider the incircle of center I and the nine points circle of Euler. The incircle is interior tangent to the Euler circle in a point called Feuerbach point.

The harmonic series is divergent

Posted June 28th, 2008 by Structure

The harmonic series is:

$$ 1 + \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+...+ \frac{1}{n}+...$$

This series is divergent.

Proof

$$ 1 + \frac{1}{2}+ (\frac{1}{3}+ \frac{1}{4})+ (\frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8})+... > $$
$$ 1 + \frac{1}{2}+ (\frac{1}{4}+ \frac{1}{4})+ (\frac{1}{8}+ \frac{1}{8}+ \frac{1}{8}+ \frac{1}{8})+... = $$
$$ 1 + \frac{1}{2}+ \frac{1}{2}+ \frac{1}{2}+ ... = +\infty $$

Projection of a point on a plane

Posted March 22nd, 2008 by Structure

Let consider the plane (p) of equation

$$(p)\:ax+by+cz+d=0$$

and a point M(u,v,w)
We look for the point $ N =(x_0,y_0,z_0) $, the projection of M on the plane.Normal of the plane is the vector (a,b,c) so line by M(u,v,w) of equations

$$\frac{x-u}{a}=\frac{y-v}{b}=\frac{z-w}{c}=t$$

is the line perpendicular on the plane.
so we have for a point on this line parametric equations

$$\left<br /> {\begin{array}{c}<br /> x=u+at\\<br /> y=v+bt\\<br /> z=w+ct<br /> \end{array}$$
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