We would like to start our expolaration with following constructions:
1) Construct any triangle ABC and construct the orthocenter of ABC
If you would like to see GSS construction of orthocenter any triangle click here
In construction of the orthocenter for HBC, the altitude from H to BC is the same altitude from A to BC. The altitude from B to HC lies on segment AB and also the altitude drawn from C to HB is also lies segment AC.The orthocenter of HBC triangle is A. In other words, A=HHBC
In construction of the orthocenter for HBA, the altitude from H to AB is the same altitude from C to AB. The altitude from B to HA lies on segment BC and also the altitude drawn from A to HB is also lies segment AC.The orthocenter of HBatriangle is C. In other words, C=HHBA
In construction of the orthocenter for HAC, the altitude from H to AC is the same altitude from B to AC. The altitude from A to HC lies on segment AB and also the altitude drawn from C to HA is also lies segment BC.The orthocenter of HAC triangle is B. In other words, B=HHAC
We observed that the orthecenters of triangles HAC, HAB and HBC are B,C and A (one of the vertexesof the original triangle ABC)
Click here to see animation ........................Click here to see animation
As we proved below the length of four circles are the same anyhow we move original ABC triangle.
EXAMINATION OF NINE-POINT CIRCLE FOR ALL TRIANGLE:
It was named by French Mathematician Jean-Victor Poncelet. Karl Feuerbach made many expolaration with nine-point circle.
The nine-point circle passes through the midpoint ma,mb and mc, and M,K,L(midpoints of AH,BH and CH)
N point( center of nine-point triangle) lies on the same line which passes through GH.
As we can see above, nine-point circle is tangent to incircle and the other circles(green,blue and red one) are tangent to the extended sides of ABC triangle. They are also tangent to the nine-point circle.
We found that the nine-point circles of the triangles,ABC,HAC,HAB and HBC are the same.
The interesting observation is : .That is true all different possible triangles.