Center of triangle, Circle

1. The Centroid (G) of a triangle is the common intersection of the three medians.

2. 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 line of the opposite side.

3. 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 of the segment determined by the two points, C is on the perpendicular bisector of each side of the triangle.

5. Circumcircle is the circumscribed circle of the triangle with the center of circumcenter.

4. 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.

5. Incircle is the inscribed circle of the triangle with the center of incenter .

6. Center of Nine point circle

7. The Nine Point Circle for any triangle passes through the three mid-points of the sides, the three feet of the altitudes, and the three mid-points of the segments from the respective vertexes to orthocenter.

8. Triangle centers (H,G,C, and I)

9. Triangle centers with Euler Line

Triangles

10. A triangle connecting the three midpoints of the sides is called the Medial triangle.

11. Take a triangle ABC. Construct Orcenter H and the segments HA, HB, and HC. Construct the midpoints of HA, HB, and HC. Connect the midpoints to form a triangle. Here is GSP file of Orthocenter , Midsegment triangle .

12. A triangle connecting the feet of the altitudes is called the Orthic triangle.

13. Pedal triangle

Divide a line segment

15. Divide a segment AB into two parts that form a golden ratio.

Regular Polygons

16. Equilateral triangle , given a side

Isosceles triangle

22. Isosceles triangle, given base and altitude

Locus and Envelop

23. Locus of vertex of a fixed angle that subtends a fixed segment.

24. Parabola 1

25. Parabola 2

26. Conics 1

27. Conics 2

Tangent circles

28. Tangent circles 1 Given two circles and a point on one of the circles. Construct a circle tangent to the two circles with one point of tangency being the designated point.

Projective geometry

Tool infomation : The Geometer's Sketchpad Version 4