Assignment 8:

Serkan Hekimoglu and Jamie Parker

EMAT 6680

University of Georgia (Dr.Wilson)

Altitudes and Orthocenters

We would like to start our expolaration with following constructions:

1) Construct any triangle ABC and construct the orthocenter of ABC

Figure 1: Construction of ABC triangle and its orthocenter.

If you would like to see GSS construction of orthocenter any triangle click here

Figure 2: Constructing the orthocenter for HBC,HAB and HAC

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

Figure 3:Construction of circumcircles of triangles ABC, HAB, HAC and HBC.

We observed that the orthecenters of triangles HAC, HAB and HBC are B,C and A (one of the vertexesof the original triangle ABC)

Figure 4: Changing A,B and C points locations

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.

Figure 6:

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.

Figure 7: nine-point cirle and triangles excirles

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.

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