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Handbook of mathematics for engineers and scienteists part 13

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A straight line through a vertex of a triangle and the midpoint of the opposite side is called a median of the triangle (Fig. 3.2a). The three medians of a triangle intersect in a single point lying strictly inside the triangle, which is called the centroid or center of gravity of the triangle. This point cuts the medians in the ratio 2:1 (counting from the corresponding vertices). | 52 Elementary Geometry One can find the area of an arbitrary polygon by dividing it into triangles. 3.1.2-2. Properties of quadrilaterals. 1. The diagonals of a convex quadrilateral meet. 2. The sum of interior angles of a convex quadrilateral is equal to 360 Fig. 3.14a and b . 3. The lengths of the sides a b c and d the diagonals d1 and d2 and the segment m connecting the midpoints of the diagonals satisfy the relation a2 b2 c2 d2 d2 d2 4m2. 4. A convex quadrilateral is circumscribed if and only if a c b d. 5. A convex quadrilateral is inscribed if and only if a 7 3 6. 6. The relation ac bd d1d2 holds for inscribed quadrilaterals Ptolemy s theorem . Ò b Figure 3.14. Quadrilaterals. 3.1.2-3. Areas of quadrilaterals. The area of a convex quadrilateral is equal to S 2d1d2 sin q yjp p - a p - b p - c p - d - abcd cos2 3.1.2.1 where p is the angle between the diagonals d1 and d2 and p 2 a b c d . The area of an inscribed quadrilateral is S a p p - a p - b p - c p - d . 3.1.2.2 The area of a circumscribed quadrilateral is S y abcdsin2 3 . 3.1.2.3 If a quadrilateral is simultaneously inscribed and circumscribed then S V abcd. 3.1.2.4 3.1.2-4. Basic quadrilaterals. 1 . A parallelogram is a quadrilateral such that both pairs of opposite sides are parallel Fig. 3.15a . 3.1. Plane Geometry 53 b a Figure 3.15. A parallelogram a and a rhombus b . Attributes of parallelograms a quadrilateral is a parallelogram if 1. Both pairs of opposite sides have equal length. 2. Both pairs of opposite angles are equal. 3. Two opposite sides are parallel and have equal length. 4. The diagonals meet and bisect each other. Properties of parallelograms 1. The diagonals meet and bisect each other. 2. Opposite sides have equal length and opposite angles are equal. 3. The diagonals and the sides satisfy the relation d2 d2 2 a2 ft2 . 4. The area of a parallelogram is S ah where h is the altitude see also Paragraph 3.1.2-3 . 2 . A rhombus is a parallelogram in which all sides are of equal length .