Question 32
In the figure, $AB$ is a fixed segment and $C$ is a variable point within it. Equilateral triangles are constructed $ACB'$
and $CBA'$ of sides $AC$ and $CB$ in the same half-plane defined by $AB$, and another equilateral triangle $ABC'$ of side $AB$ in the opposite half-plane. Demonstrate:
a) The lines $AA'$, $BB'$ and $CC'$ are concurrent.
b) If we call $P$ the point common to the three lines in part a), find the locus of $P$ when $C$ varies in the segment $AB$.
c) The centers $A''$, $B''$ and $C''$ of the three triangles form an equilateral triangle.
d) Points $A''$, $B''$, $C''$ and $P$ are concyclical.
and $CBA'$ of sides $AC$ and $CB$ in the same half-plane defined by $AB$, and another equilateral triangle $ABC'$ of side $AB$ in the opposite half-plane. Demonstrate:
a) The lines $AA'$, $BB'$ and $CC'$ are concurrent.
b) If we call $P$ the point common to the three lines in part a), find the locus of $P$ when $C$ varies in the segment $AB$.
c) The centers $A''$, $B''$ and $C''$ of the three triangles form an equilateral triangle.
d) Points $A''$, $B''$, $C''$ and $P$ are concyclical.