Geometric Constructions 11

A wooden
triangle moves on a plane so that two vertices move along two fixed perpendicular
lines. How does the third vertex move?

Generalized
the above situation to two fixed lines not necessarily perpendicular to
each other.

Illustrate
Napoleon's
Theorem: the centers L, M, N of the three equilateral triangles
DBXC,
DCYA,
DAZB
built outwards on the sides BC, CA, AB of an arbitrary triangle DABC
are the vertices of an equilateral triangle. The same is true of the centers
of the three inward equilateral triangles.

Generalize
Napoleon's Theorem.

Explore
interesting properties associated with the Torricelli's configuration.

Investigate
the properties of the figure consisting of a quadrilateral together with
the four squares erected outwardly on the sides.

Design
a figure to illustrate the following cases of indeterminate forms
all at the same time: