Projection of vector

edit

Your approach will not work since you try to play with plots. Instead, you have to play with vectors, not their plots.

To my understanding of the picture, D is the orthogonal projection of A to the line (B,C).

If v1 denotes the vector BA and v2 denotes the vector BC, the length j is the inner product of v1 by v2 divided by the norm of v2. Here is how to do it in Sage:

sage: A = (9, 5); B = (2, 4); C = (16, -2);
sage: v1 = vector(A)-vector(B)
sage: v2 = vector(C)-vector(B)
sage: v1.inner_product(v2)/v2.norm()
23/29*sqrt(58)

If you want a floating approximation, you can do:

sage: RDF(v1.inner_product(v2)/v2.norm())
6.040095911547238

EDIT : When you write

sage: plot(x,(x,2,5),color='red')

You ask Sage to plot the function x (i.e. the identity function), for x varying from 2 to 5.

To compute the coordinates of D, you can do:

sage: BA = -AB
sage: D = vector(B) + BC*BA.inner_product(BC)/BC.norm()/BC.norm()
sage: D
(219/29, 47/29)

To add the point D on the picture, you can do:

sage: P += point(D)

If you want the x-axis and the y-axis to have the same scale (so that the right angles are visible), you can do:

sage: P.set_aspect_ratio(1)