Centroid - The centroid of a triangle is the intersection of its medians. This center is also called the center of gravity because this is the point at which the entire triangle can be balanced.

Orthocenter - The orthocenter of a triangle is the intersection of the perpendicular bisectors of the sides of the triangle.

Circumcenter - The circumcenter is the point which is equidistant to all three vertices of the triangle.

Circumcircle - The circumcircle is circle that is cricumscribed about a triangle.

Incenter - The incenter is the intersection of the angle bisectors of a triangle.

Incircle - The incircle is the circle inscribed inside a triangle.

Medial Triangle - The medial triangle is the triangle formed using the midpoints of the sides of the original triangle as the vertices.

Orthocenter, Mid-segment Triangle - The orthocenter of the mid-segment triangle is the orthocenter of the medial triangle and can be found without constructing the medial triangle.

Orthic Triangle - The orthic triangle is the triangle with the vertices being the feet of the altitudes of the original triangle.

Pedal Triangle - The pedal triangle is the triangle formed using any point p and constructing the perpendiculars to the sides of the original triangle. The points of intersection are the vertices of the pedal triangle.

Center of Nine Point Circle - The center of the nine point circle can be found simply by finding the circumcenter of the medial triangle, because it only takes three points to define a circle.

Nine Point Circle - The nine point circle is the circle on which lie the feet of the altitudes, the midpoints of the sides, and the midpoints of the segments connecting the orthocenter to the vertices.

Trisectiong a Line Segment - This is done using similar triangles.

Equilateral Triangle, given a side - This construction involves circles and intersections.

Square, given a side - This construction utilizes circles and perpendicular lines.

Isosceles Triangle, given base and altitude - This construction uses the Isosceles Triangle Theorem, which involves the perpendicular bisector of the base.

Triangle Centers (H, G, C, and I) - H is the orthocenter, G is the centroid, C is the circumcenter, and I is the incenter. These are be constructed separately above.

Triangle Centers with Euler Line - The Euler Line is the line segment connecting H, G, and C.

Construct the point which divides segment AB into the Golden Ratio - This can be constructed using circles and diagonals.

Find the mean porportional of segments AB and BC - The mean porportional segment is the segment m, whose length squared is equal to (AB)*(BC).

Rhombus, given a side and an altitude - This construction uses parallel lines, circles, and the definition of altitude.

Pentagon, given a side - This construction creates a regular pentagon.

Pentagon, given a radius - This construction creates a regular pentagon using a circle with a given radius.

Hexagon, given a side - This construction creates a regular hexagon.

Octagon, given a side - This construction creates a regualr octagon.