Problem:[br]Inscribe an equilateral triangle [math]DEF[/math] in a given triangle [math]ABC[/math].[br][br]Solutions:[br]For any equilateral triangle a rotation with center one of the vertices and angle [math]60^{\circ} [/math] will map one of the remaining vertices onto the third vertex.[br]Let point [math]D[/math] be an arbitrary point on segment [math]AB[/math]. [br][br][list][br][*]Drag the slider to the end to rotate segment [math]BC[/math] on angle [math]60^{\circ} [/math] around point [math]D[/math].[br][/list][br]If vertex [math]F[/math] is on segment [math]BC[/math], then the third vertex [math] E [/math] should be on the image [math]B’C’[/math] of [math]BC[/math], but it also has to be on the segment [math]AC[/math]. Therefore, vertex [math]E[/math] is the intersection point of the segments [math]AC[/math] and [math]B’C’[/math]. This determines the side of the equilateral triangle. [br][list][br][*]Click on the Construction button to finish the construction.[br][*]When this problem has a solution?[br][*]Drag point D on segment AB. Notice the relation between the intersecting points of the circle and the position of D.[br][*]Drag the vertices of the triangle. Construct triangles for which the circle and segment BC have one intersection point. [br][/list][br][br]A geometric construction using this transformation was first described by I. M. Yaglom, in [i]Geometric Transformations I[/i], MAA, 1962, Chapter 2, Problem 18