Calculating the length of a line in a triangle

Question

I feel very stupid, but I have to answer this question but I cannot seem to solve it! :( I have to find the length of DF. I already figured out that because angle C = angle A1 (left part of the angle) Thales applies so AD must be 12 as well. But I don't know how to proceed. Anyone has a clue? Thanks in advance!

Answer 1

F is the intersection of all three medians of the triangle. You can know the length of AD (which is a median) with http://en.wikipedia.org/wiki/Apollonius%27_theorem if you have the length of the 3 sides (in ABC), or http://en.wikipedia.org/wiki/Law_of_cosines if you have 2 angles and 1 length (in ACD), which you seem to have. After that, it is a result that the length of DF is equal to the third of the length of AD.

EDIT : I just saw that ABC is rectangle. That simplifies the problem of knowing the length of AD if you know that if ABC is rectangle in A, BC is the diameter of a circle passing by A, B and C. What relation can you deduce between AD and BD ?

Answer 2

If you know $AD = 12$ already, then $DF = 12/3$. This is because $F$ is the centroid of $\triangle ABC$, and the centroid of a triangle divides each median into $1:2$.