Construct any triangle ABC and Orthocenter H of triangle ABC. Construct the Orthocenters of triangles HBC, HAB, and HAC. Click Here for a GSP file .

We can find the orthocenters HBC, HAB, and HAC are on the opposite vertexes of the triangle ABC. Why?

Construct altitudes of the triangle ABC. Let'ts try to explain it. Click Here for GSP file.

Construct Circumcenters with Circumcircles of triangles ABC, HBC, HAB, and HAC. Click Here for GSP file.

We can find radiums of the four Circumcircles are same.

Construct the nine points circles for triangles ABC,HBC,HAC,and HAB. Click Here for GSP file.

We can find the four nine points circles are coincidence.

Construct triangle ABC, its incircle, its three excircles, and its nine-point circle.

Nine points circle tangent the incircle and the three excircles.