By Choquet-Bruhat Y., DeWitt-Morette C., Dillard-Bleick M.

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**Sample text**

If this curve is viewed from the side of its plane opposite to that on which 0 lies, the order E1E2E3 defines a sense of circulation. If this sense is counterclockwise, the set e1, e2, e3 is said to be right-handed or dextral; for it is then possible to extend the thumb, index, and middle fingers of the right hand so that they have the directions of e1, e2, e3, BASE VECTORS § 12 2'r respectively. If the sense defined by E1E2E3 is clockwise the set is said to be left-handed or sinistral. Any vector u = PQ may be expressed as the sum of three vectors parallel to el, e2, e3, respectively.

Center of a triangle lies on the line joining the orthocenter to the circumcenter and divides it in the ratio of 2/1. This line is called the Euler line of the triangle. Example 5. In order to interpret the identity, (a + b - c - d)2- (a - b - c+ d)2 = 4(a - c) - (b-d), with reference to the plane quadrilateral ABCD, let P, Q, R, S denote the mid-points of AB, BC, CD, DA; then (Fig. 15e) we have (PR)2 - (QS)2 = CA DB. Hence, if the diagonals of a quadrilateral cut at right angles, the lines joining the mid-points of opposite sides are equal.

If we regard a, b, c, d as the position vectors of the vertices of a space quadrilateral ABCD, p = (a + c), q = (b + d) locate the mid-points P, Q of the diagonals AC, BD; 2 hence (AB)2 + (BC)2 + (CD)2 + (DA)2 = (AC)2 + (BD)2 + 4(QP)2. The sum of the squares of the sides of any space quadrilateral equals the sum of the squares of its diagonals plus four C times the square of the segment joining their middle points. Example 4. The identity, (a-b) (h-c) 0, B Fia. 15b shows that the altitudes of a triangle ABC meet in a point H, the ortho- center of the triangle (Fig.

### Analysis, Manifolds and Physics. Basics by Choquet-Bruhat Y., DeWitt-Morette C., Dillard-Bleick M.

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