In the plane there are given the lines $\ell_1$, $\ell_2$, the circle $\mathcal{C}$ with its center on the line $\ell_1$ and a second circle $\mathcal{C}_1$ which is tangent to $\ell_1$, $\ell_2$ and $\mathcal{C}$. Find the locus of the tangent point between $\mathcal{C}$ and $\mathcal{C}_1$ while the center of $\mathcal{C}$ is variable on $\ell_1$. Mircea Becheanu
Problem
Source: Romanian IMO Team Selection Test TST 1988, problem 7
Tags: geometry proposed, geometry
S.E.Louridas
10.08.2010 15:25
My opinion (generally): Let O be the section point of $\ell _1 ,\ell _2 ,OX$ the bisector of the angle of $\ell _1 ,\ell _2 .$ Consider $C_1 $ instantaneously constant witch tangents $\ell _1 \;at\;T$ and intersects the bisector at P, every point of the minor arcTP is a tangent point one circle 'type' $C $ (a family of circles...) with $C_1$. Finally the locus is the union of the angle <XOD with D point of $l_1$ and the symmetric of <XOD with respect to the point O (and the point O of course). S.E.Louridas
S.E.Louridas
10.08.2010 18:49
Sorry, I forgot to mention that we work the same way for the other (π-φ) of the angle is being formed by $\ell _1 ,\ell _2 \;also..$ so we have an other set (respectively) of points of focus S.E.Louridas
S.E.Louridas
16.08.2010 14:29
S.E.Louridas wrote: Sorry, I forgot to mention that we work the same way for the other (π-φ) of the angle is being formed by $\ell _1 ,\ell _2 \;also..$ so we have an other set (respectively) of points of focus. Finally finally The locus is the whole plane while it is sufficient to consider the point us a circle with radius 0. S.E.Louridas