Pythagoras’s theorem

If and r are the radii of the cation and anion respectively then by applying Pythagoras s theorem to triangle ABC we find that... 

Next, we have 12 Na ions (the ones located on the faces of the cubic unit cell). Using Pythagoras s theorem we can show that they are at the distance of d-Jz from the same Na we looked at above (see Figure T3.2 below). This is a repulsive interaction for which we get the following potential Fj ... 

Next to consider are eight CI ions at the middle of each edge of the unit cell. Again, using Pythagoras s theorem we can see that the distance between them and our Na is. The potential F3 is thus ... 

The most commonly used distance metric is Euclidean distance. The Euclidean distance between two points is the everyday distance we measure with a ruler or measuring tape. It can be calculated easily using Pythagoras s theorem and is the square root of the sum of the square distances between the points in each dimension. 

If we apply Pythagoras s theorem of plane geometry to the right-angled triangle in Figure 7.2, we have [r2 = (D + x)2 + PRin(,], and inserting this expression into Equation (525) we obtain... 

Further, applying Pythagoras s theorem to triangle BCE in Figure 15.3 produces an expression for the magnitude of the inlet velocity relative to the blades, Cri. 

Applying Pythagoras s theorem to the triangle AFD then gives the magnitude of the outlet velocity, ca, as ... 

Let the two points have Cartesian coordinates (jci,ti) and (x2,T2). The element of distance, ds, along a varied path, such as that shown in Fig. AII.l, is given by Pythagoras s theorem as... 

Since all tetrahedron edges are equal, the value of z can be obtained as a function of x. Applying Pythagoras s theorem to the appropriate coordinates leads to ... 

Using Pythagoras s theorem, the relationship between fractional coordinates for oxygen atom (x,y,z) and cation at X, X, Xc can be derived ... 

FIGURE 4.2 Einstein s proof of Pythagoras theorem. The proof does assume, perhaps prematurely, that the angles of a triangle add up to 180°. 

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