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.