TAILIEUCHUNG - Handbook of mathematics for engineers and scienteists part 24

In a triangle, the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the segments of the altitudes between the orthocenter and the vertices all lie on a single circle, the nine-point circle (Fig. ). | . Line and Plane in Space 129 Figure . Plane passing through a point and parallel to two straight lines. . Equation of plane passing through point and parallel to two straight lines. The equation of the plane passing through a point M1 x1 y1 z1 and parallel to two straight lines with direction vectors Ri 1 m1 n1 and R2 2 m2 n2 see Fig. is x - x1 y - y1 z - z1 11 m1 1 12 m2 2 0 or r - ri RtR 0 where r and r1 are the position vectors of the points M x y z and M1 x1 y1 z1 respectively. Example 7. Let us find the equation of the plane passing through the point M1 0 1 0 and parallel to the straight lines with direction vectors R1 1 0 1 and R2 0 1 2 . According to the desired equation is x - 0 y - 1 z - 0 1 0 1 0 1 2 0 whence -x - 2y z 2 0. . Plane passing through two points and perpendicular to given plane. The plane see Fig. passing through two points M1 x1 y1 z1 and M2 x2 y2 Z2 and perpendicular to the plane given by the equation Ax By Cz D 0 is determined by the equation x - x1 x2 - x1 A y - y1 y2 - y1 B z - z1 Z2 - z1 C 0 or r - r1 r2 - n N 0 where r r1 and r2 are the position vectors of the points M x y z M1 x1 y1 z1 and M2 x2 y2 z2 respectively. Remark. If the straight line passing through points M1 x1 y1 z1 and M2 x2 y2 z2 is perpendicular to the original plane then the desired plane is undetermined and equations become identities. 130 Analytic Geometry Figure . Plane passing through two points and perpendicular to given plane. Example 8. Let us find an equation of the plane passing through the points M1 0 1 2 and M2 2 2 3 and perpendicular to the plane x - y z 5 0. According to the desired equation is x - 0 y - 1 z - 2 2 - 0 2 - 1 3 - 2 1 -1 1 0 whence 2x - y - 3z 7 0. . Plane passing through point and perpendicular to two planes. The plane see Fig. passing through a point M1 x1 y1 z1 and perpendicular to two nonparallel planes A1x B1y C1z D1 0 and A2x B2y C2Z D2 0 is .

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