Symmedians (Cevian)

Symmedians

A symmedian is the reflection of a median of a triangle about its corresponding angle bisector. For example, in the triangle below, the blue lines are medians, the green lines are angle bisectors, and the red lines are symmedians:

As you can see from the diagram, the symmedians are just another type of cevian in the triangle. Also, note that by its definition, the symmedians are isogonal lines to the medians.

We will now look at a well known result on symmedians.

Suppose we are given triangle $\triangle ABC$. Draw the circumcircle of $\triangle ABC$ and draw the tangents through $A$ and $B$. Let the intersection of the tangents be $X$. Then $AX$ is the symmedian of $\triangle ABC$ through $D$.

Proof:

Now we use this fact to prove that the symmedians are concurrent.

Proof:

Notice that the circumcircle is the incircle of $DEF$. Also, we know that $AD$, $BE$, $CF$ intersect at the Gergonne point of $DEF$ by Ceva's theorem. Thus, we conclude that the symmedians are concurrent. $\Box$

That's the basics of symmedians. Thanks for reading!