CENTROID

The center of gravity of a triangle, where the three medians of the triangle intersect

ORTHOCENTER

The orthocenter of a triangle is the point
where all three of its altitudes intersect.

CIRCUMCENTER

The point where the three perpendicular bisectors of the
sides of a triangle meet

CIRCUMCIRCLE

The circumcircle or circumscribed circle
of a polygon is a circle which passes through all the
vertices of the polygon. The center of this circle is called the circumcenter.

INCENTER

The incenter of a triangle or regular
polygon is the point where the angle bisectors meet.

INCIRCLE

It is the largest circle that will fit
inside the triangle.

MEDIAL TRIANGLE

It
is the small triangle with vertices at the midpoints of a bigger triangle

MIDSEGMENT TRIANGLE
ORTHOCENTER

A midsegment of a triangle is a segment connecting the
midpoints of two sides of a triangle.

ORTHIC TRIANGLE

The triangle joining the feet of the
altitudes of a triangle is called the orthic triangle. 
PEDAL TRIANGLE

The pedal triangle is the triangle
whose vertices are the feet of the perpendiculars from the point to the sides
of the given triangle. 
NINE POINT CIRCLE

It is a circle that
passes through nine significant concyclic
points defined from the triangle 
CENTER OF THE NINE
POINT CIRCLE

The center of the circle defined above 
TRISECTING A LINE
SEGMENT

Dividing a segment into three equal
parts 
EQUILATERAL TRIANGLE
GIVEN ONE SIDE

Equilateral triangles have equal sides
and angles. Here one is constructed given one side. 
SQUARE GIVEN A SIDE

Four equal sides and four ninety
degree angles define a square. Here one is constructed given a side. 
ISOSCELES TRIANGLE
GIVEN BASE AND ALTITUDE

Isosceles triangles have two equal
sides and two equal angles. Here one is constructed given a base and altitude 
TRIANGLE CENTERS

Orthocenter, Centroid, Circumcenter
and Incenter 
TRIANGLE CENTERS WITH EULER LINES

It is a line that passes through several important points determined
from the triangle, including the orthocenter, the circumcenter, the centroid, and the center of the ninepoint
circle of the triangle. 
LOCUS VERTEX OF A FIXED ANGLE SUBTENDING A FIXED VERTEX

Here one is constructed 
DIVIDE A SEGMENT AB INTO TWO PARTS THAT FORM A GOLDEN RATION

Here one is constructed 
PENTAGON GIVEN A RADIUS 
A polygon with five sides and five
angles. Here one is constructed given a radius 
PENTAGON GIVEN A SIDE

A polygon with five sides and five
angles. Here one is constructed given a side 
HEXAGON GIVEN A SIDE

A polygon with six sides and six
angles. Here one is drawn given a side 
OCTAGON GIVEN A SIDE 
A polygon with eight sides and eight
angles. Here one is drawn given a side 