Write-up 8

Altitudes and Orthocenters: A further look at the construction of 9-point circles

T. Barron

Follow along with the constructions as indicated below. If you have access to GSP, you can click on the blue highlighted part of the instructions to see a detailed GSP script. You can also link to a GSP construction of the following pictures by clicking on the specified highlighted blue words.

1) Construct any triangle ABC. Click here for a GSP construction and here for a GSP script for constructing a triangle.

 

Figure 1

 

2) Construct the Orthocenter H of triangle ABC. Click here for a GSP construction and here for a GSP script.

Figure 2

 

3) Construct the Orthocenter of triangle HBC. Click here for a GSP construction and here for a GSP script. Notice that the Orthocenter of triangle HBC is point A.

Figure 3

 

4) Construct the Orthocenter of triangle HAB. Click here for a GSP construction and here for a GSP script. Notice that the Orthocenter of triangle HAB is point C.

Figure 4

 

5) Construct the Orthocenter of triangle HAC. Click here for a GSP construction and here for a GSP script. Notice that the Orthocenter of triangle HAC is point B.

Figure 5

 

6) Construct the circumcircles of triangles ABC, HBC, HAB, and HAC. Click here for a GSP construction and here for a GSP script.

Figure 6

Thus, because of the defining characteristics of the nine-point circle as discussed previously, it should be clear that the nine-point circle construction for triangles ABC, HBC, HAC, and HAB should all be the same circle as shown below. Click here to manipulate the nine-point circle as indicated below.

Figure 7

 

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