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Around three points on a side create equilateral triangle

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khongghen

khongghen

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Around three points on a side create equilateral triangle
 

1. Problem 1: Let triangle ABC, AP=AB/2; angle BAP =60 degree; AN=AC/2 angle NAC=60 degree. M1 is midpoint of BC => M1NP is equilateral triangle. Figure 1
Figure 1: M1NP is an equilateral triangle
2. Problem 2: M1N meet AB at Q, M1P meet AC at K. M1QK is an equilateral triangle. Figure 2
Figure 2: M1QK is an equilateral triangle
3. Problem 3: A,P,K,Q,N are concyclic. Figure 3
Figure 3: A,N,Q,K,P are concyclic
4-Problem 4: We call Circle through A,N,Q,K,P is circle matched with A. Circle matched with A,B,C concurrent at D. Figure 4
Figure 4: Circle respond with A,B,C concurrent at D
5-Problem 5: H,G,I are perpendicular D to AB,BC,AC. HGI is an equilateral triangle. Figure 5
Figure 5: HGI is an equilateral triangle
6-Problem 6: O1,O2,O3 are circumcircle of circle matched with A,B,C. M1,M2,M3 are midpoints of AB,BC,CA. O1M1,O2M2,O3M3 are concurrent. Figure 6.
Figure 6: O1M1,O2M2,O3M3 are concurrent
7-Problem 7: AO1,BO2,CO3 are concurrent. Figure 7.
8-Problem 8: We call the triangle M1NP at item 1 is a equilateral triangle matched A. Centroid of equilateral triangle matched A,B,C is equilateral triangle.

Around three points on a side create an equilater triangle.JPG

 






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