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1) Take any quadrilateral. Let A, B, C and D be its vertices. Let P, Q, R and S be the middle points of its edges. Show that the quadrilateral PQRS is a parallelogram.

Hint: Show that vector SP= vector RQ and that vector SR= vector PQ

2) We consider two spheres. The sphere of radius 3 centered at the origin (0, 0, 0). The second sphere is tangent to the first sphere and is centered at A(3, 5, 2). Find the point of intersection P of these spheres.

Hint: The points A, P and the origin are on the same line.

1. From your figure, consider the points A, D and C and the distance between points A and C = |AC|. Using the relationships for similar triangles, the distance between the mid-points S and B is |SB| = (1/2)|AC| and SB is paralell to AC. SImilarly, the line segment |PQ| = |SB| and these 2 segments are paralell.

Using the same approach, it can be shown that the segments SP and RQ are also equal and paralell. The 4 segments together thus form a paralellogram.

2. As stated in the hint, the points O, P and A are on a line (sort of by definition). The distance to the point P along the x, y and z axes will be in the same ratio to the coordinates of A = (3, 5, 2) as is the radius of 3 is to the distance to A. The distance to A is sqrt(3^2 + 5^2 + 2^2) = D, by the Pythagorean Thm. So we have the ratio R = 3/D and the coordinates of P are (3R, 5R, 2R).

Randy

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