Prove that Triangle ABC is the medial triangle of Triangle A

Prove that Triangle ABC is the medial triangle of Triangle A\"B\"C\"

Solution

Solution:

There are two medians of ABC, namely AA\' and BB\', intersect two sides of A\'B\'C\', namely C\'B\' and C\'A\', in points D and E respectively.

We need to show that D is the midpoint of C\'B\' and E is the midpoint of A\'C\'. We know that C\'B\' // BC and AC\'/AB = 1/2. So, AC\'D is similar to ABA\' and C\'D = BA\'/2. Similarly, DB\' = A\'C/2. Since A\' is the midpoint of BC, BA\' = A\'C. Hence C\'D=DB\' and D is the midpoint of C\'B\'. With a similar argument E is the midpoint of A\'C\'. Therefore, G which is the centroid of ABC, is also the centroid of A\'B\'C\'.

Hence it proves that the  Triangle ABC is the medial triangle of Triangle A\"B\"C\"