Hi, I want a help to solve a geometry questions and I hope you give me solutions of them as soon as possible within two day.......thanks

1.Let B an affine plane, and let i,j be a basis for the associated vector space P . let A and B be the point whose position vector relative to some fixed origin O, are 3i+2j and 6i+8j. find the position vector of the point C on AB such that AC:CB =2:1. find also the position vector of the point D on that line such that C and D divide AB harmonically.

2.Three points in the plane P of the previous question have of positions vectors x1i + y1j, x2i+y2j, and x3i+y3j. prove that they are collinear if and only in the determinant
1 x1 y1
1 x2 y2
1 x3 y3
is equal zero.

3.According to Ceva's Theorm, the medians of triangle ABC all meet on a point. Find the position vector of this point (called the centroid) in terms of the position vectors A, B and C.

4.Let L and M be lines in some affine plane and let O be a point on either line. Define a projection from O to be operation which takes a point X on L to the unique point Y on M such that O, X and Y are collinear.
Prove that if x and y are the coordinates of X and Y relative to fixed rulers on L and M then
y=ax+b/cx+d
where a,b,c and d are constant.

5.Show that circle and a line can not meet at more than two point.


( HINT: question from book: Elementary Geometry: John Roe, Oxford Science Publication, 1993)

 

 

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