GeometricLocusVertex2::usage="Let A, B be two fixed points and C a variable point on the X axis. Draw the line AC and call D the point of intersection of this line with the axis Y. Draw the line ℒ parallel to AB by the point D. Draw the line that connects C with B and call M the point where this line intersects with ℒ. Find the geometric locus generated by M as C moves. References: Carrillo Hoyo, Ángel et al. Lugares Geométricos. Instituto de Matemáticas, Facultad de Ciencias, UNAM; AI Assistant; Personal.";
Animation:
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Animate[A={2,2};B={5,3};xc[t_]:=4+1.5Cos[t];yd[t_]:=A[[2]]/(1-A[[1]]/xc[t]);D1[t_]:={0,yd[t]};C1[t_]:={xc[t],0};M[t_]:=Module[{u,v,sol},sol=Solve[D1[t]+u(B-A)==B+v(C1[t]-B),{u,v},Reals];D1[t]+u(B-A)/.First[sol]];Show[{ParametricPlot[M[t],{t,0,t1},ColorFunction"BlueGreenYellow"],Graphics[{Thick,Blue,Line[{A,B}],Red,Line[{D1[t1],M[t1]}],Black,Line[{D1[t1],C1[t1]}],Black,Line[{C1[t1],M[t1]}],Black,PointSize[0.02],Point[A],Text[Style["A",14,Bold],A,{0,-1.2}],Point[B],Text[Style["B",14,Bold],B,{0,-1.2}],Point[D1[t1]],Text[Style["D",14,Bold],D1[t1],{0,-1.2}],Point[C1[t1]],Text[Style["C",14,Bold],C1[t1],{0,-1.2}],Point[M[t1]],Text[Style["M",14,Bold],M[t1],{0,-1.2}]}]},Axes->True,AspectRatio->Automatic,AxesLabel->{x,y},AxesOrigin->{0,0},PlotRange->{{0,15},{0,15}},ImageSize->450],{t1,0.00001,2π,0.00001},AnimationRunning->False]
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Note: How can you see the geometric locus of M is a straight line.