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1. Centroid: The Centroid is the point of concurrence of the 3 medians (segment from vertex to midpoint of opposite side).
2. Orthocenter: The Orthocenter is the point of concurrence of the three lines containing the altitudes of the triangle.
3. Circumcenter: The Circumcenter (C) is the point in the plane equidistant from the 3 vertices. Therefore C lies on the perpendicular bisectors of each side of the triangle.
4. Circumcircle: The Circumcircle is the circle containing the vertices of a triangle. Its center is the Circumcenter of the triangle
5. Incenter: The Incenter (I) is the point on the interior of the triangle equidistant from the 3 sides. I is the point of coincidence of the angle bisectors of the 3 angles of the triangle
6. Incircle: The Incircle is the inscribed circle of the triangle. The incenter is the center of the Incircle.
7. Medial Triangle 1: The Medial Triangle is formed by connecting the midpoints of the sides of a triangle.
7a. Medial Triangle 2: Medial Triangle is constructed independent of the original triangle
7b. Orthocenter, Mid-segment triangle: The triangle formed by connecting the midpoints of the segments between the vertices and the orthocenter of a triangle.
8. Orthic triangle: In an acute triangle, the Orthic triangle is formed by connecting the feet of the altitudes
9. Pedal triangle: Given any triangle ABC, locate any point P in the plane. Form a triangle by constructing perpendiculars to the sides of ABC (extended if necessary) locate three points R, S, and T that are the intersections. Triangle RST is the Pedal Triangle.
10. Center of Nine point circle:
12. Nine Point Circle: The Nine Point Circle is formed from the midpoints of the sides of a triangle, the feet of the altitudes to each side, and the 3 midpoints of the segment from the vertices to the orthocenter.
13. Trisecting a line segment
14. Equilateral triangle, given a side
15. Square, given a side
16. Isosceles triangle, given base and altitude
17. Triangle Centers (H, G, C, and I): Orthocenter (H), Centroid (G), Circumcenter, and Incenter.
18. Triangle Centers with Euler Line: (line through Orthocenter, Centroid, and Circumcenter)
19. Sierpinski's Triangle: Fractal design created by multiple iterations of the medial triangle of an equilateral triangle
20. Angle Bisector of Disjoint Segments:
21. Half the Area of a Triangle: Given any triangle, construct a segment parallel to the base that divides the triangle into two equal areas.
22. Hidden Treasure:
23. Pentagon given radius: (needs work)
24. Square given radius: