Question

Let $H$ be the orthocenter of the equilateral triangle $\triangle ABC$. We know the distance between the orthocenters of $\triangle AHC$ and $\triangle BHC$ is $12$. What is the distance between the circumcenters of $\triangle AHC$ and $\triangle BHC$?

Answer 1

The distance between the circumcenters of triangles AHC and BHC is 8 units.

Step-by-step explanation:

To find the distance between the circumcenters of triangles AHC and BHC, we first need to understand the relationship between the centroid (G), orthocenter (H), and circumcenter (O) of an equilateral triangle.

In an equilateral triangle, G is the centroid and H is the orthocenter. The distance between G and H is 2/3 times the distance between G and O.

Since the distance between the orthocenters of AHC and BHC is given as 12, we can deduce that the distance between the circumcenters is 12 x (2/3) = 8 units.