ABC is a triangle. D is the center of BC . AC is perpendicular to AD. prove that $cos(A)*cos(c)=2*(c^2-a^2)/3$

Question

ABC is a triangle. D is the center of BC . AC is perpendicular to AD. prove that
cos(A)⋅cos(C)=2(c^2−a^2)/3ac

Answer 1

This diagram shows what a possible construction would look like:

 

https://www.desmos.com/calculator/qdrbrtf1xm

 

Looks like the intention is to use the Law of Cosines on a couple of the triangles and use the fact that BD = DC. The assumption is that the standard notation is being used (in triangle ABC, a is opp A, b is opp B, c is opp C). Try writing down the Law of Cosines for a couple of the triangles and see if you can transform the results into the desired outcome. That right angle gives you the opportunity to establish some relationships using the Pythagorean theorem.