Properties of a mid-point triangle


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

  • Circumcenter
  • Orthocenter
  • Centroid
  • None


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.

Get it on Google Play