Finding An Isosceles Triangle

Question

Can we always find an isosceles triangle of the same area as the original scalene triangle? If so, could you please provide a compass straight edge proof?

Answer 1

Draw a line parallel to the base through the third vertex of the triangle, and draw the perpendicular bisector of the base to meet the parallel line. This point is now the third vertex of the isosceles triangle whose area is the same as the original.

Answer 2

Sketch of a construction: Let the original triangle be $\Delta ABC$ and let $h$ be the height from $C$. Then of course the area is $\frac 12 h\,\overline {AB}$. Let $M$ be the midpoint of segment $AB$. Construct the perpendicular bisector, $\mathscr P$, to $AB$. Draw the line, $\mathscr L$ parallel to $AB$ through $C$ and let $X$ be the intersection of $\mathscr L$ and $\mathscr P$. Then the triangle $\Delta ABX$ does the job.