If a triangle ABC remains always similar to a given triangle,and if the point B always move along a given straight line, find the locus of C.

Question

If a triangle ABC remains always similar to a given triangle,and if the point A is fixed andB always move along a given straight line, find the locus of C.

I have been able to solve it for a right angled triangle but not a normal one.How to do this?Solve using coordinate geometry.i have already done with geometrical proof.

Answer 1

To get from B to C you always have to make a rotation of angle $\pm\theta=\angle ABC$ around $A$ and a homothety of ratio $AC/AB$ of center $A$. Both these transformations map lines to lines, so if $B$ lies on a line, the locus of $C$ is also a line, which you can find by performing the above transformations.

If you don't want, you do not need to use geometric transformations. Consider $B, B'$ two positions for $B$ and the corresponding $C,C'$ Then you can prove that the triangles $ABB'$ and $ACC'$ are similar, and from there it is quite straight forward.

Answer 2

Sounds to me like $C$ can be anywhere in the plane. For every possible $B$ on the line and every $C$ anywhere, you can always draw the line $BC$ and find a place to put $A$ such that $ABC$ is similar to the given triangle.