Question

Prove that the medians of a triangle are concurrent using triangle similarity

Answer 1

Answer:

Proof: Given triangle ABC and medians AE, BD and CF. So F is the midpoint of AB, E is the mipoint of BC and D is the midpoint of AC by definition of the median.

 

First, draw medians AE and BD and segment DE.

Claim: Triangle ABC is similar to triangle DEC.

Proof: Angle ACB = angle DCE; AC = 2CD; BC = 2CE; so similar by Side-Angle-Side Similarity Theorem.

Claim: DE//AB

Proof: Angle CDE = Angle CAB and Angle CED = Angle CBA from similarity of triangles ACE and DCE.

Claim: Angle GED = angle GAB and angle GDE = angle GBA.

Proof: DE//AB

Claim: Angle DGE = angle AGB

Proof : Vertical interior angles are congruent.

Thus triangle ABG is similar to triangle EDG by the Angle-Angle-Angle Similarity Theorem.