Properties of a mid-point triangle

Question:

!!DeltaDEF!! is the midpoint triangle of a triangle !!DeltaABC!!. Then both have the same:

  • Circumcenter
  • Orthocenter
  • Centroid
  • None

Solution

We will start with the centroid since it most easy to find out.

The centroid of a triangle !!A(x_1,y_1)!!, !!B(x_2,y_2)!!, !!C(x_3,y_3)!! is given at !!((x_1+x_2+x_3)/3,(y_1+y_2+y_3)/3)!!.

The coordinates of !!D,E,F!! are !!D((x_1+x_2)/2,(y_1+y_2)/2)!!, !!E((x_2+x_3)/2,(y_2+y_3)/2)!! and !!F((x_3+x_1)/2,(y_3+y_1)/2)!!

The centroid of even this triangle can be verified as !!((x_1+x_2+x_3)/3,(y_1+y_2+y_3)/3)!!.

Now we disprove others by considering a right angled triangle, where !!angleA!! is !!90^@!!.

Then circumcenter is point !!E!!. While for triangle !!DEF!! the mid point of !!DF!! is the circumcenter.

The orthocenter is !!A!!, while for triangle !!DEF!!. !!E!! is the orthocenter.

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