The Centroid, G, of a triangle is the common intersection of three meedians. A median of a triangle is the segment from a vertex to the midpoint of the side of the triangle opposite that vertex.
The Orthocenter H of a triangle is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the side opposite.
Let Triangle ABC be any triangle. Then if P is any point in the plane. Constuct perpendiculars to the sides of ABC (extended if necassary). Let R,S, T be the three points where the the perpendiculars and the sides (or extended sides) of ABC meet. RST is the Pedal Triangle defined by P, the Pedal Point, and triangle ABC.
The Circumcenter, C, of a triangle is the point in the plane equidistant from the three vertices of the triangle. Since a point equidistant from two points lies on the perpendicular bisector determined by the of the two points,C, is on the perpendicular bisector of each side the triangle. C may be outside the triangle.
The Circumcircle of a triangle has center at the circumcenter of the triangle and all three vertices are on the circle.
The Incenter,I, of a triangle is the point on the interior of the triangle that is equidistant from the three sides. Since a point interior to an angle that is equidistant from the two sides of the angle lies on the angle bisector, then I must be on the angle bisector of each angle of the triangle.
The Medial Triangle of a triangle ABC is the triangle constructed by connecting the three midpoints of triangle ABC. It is similar to ABC and one-fourth of its area.
The Orthhic triangle of a given acute triangle is the triangle constructed by connecting the feet of the altitudes of the given triangle.
The Nine Point Circle of a given triangle is the circle that passes through the three mid-pointss of the sides, the three feet of the altitudes, and the three midpoints of the segments from the respective vertices to the orthocenter.
The Euler line of a triangle is the segment containing H,G,C of the triangle