3D Geometry concurrency problem

Question

$ABCD$ is a tetrahedron. Let $K$ be the center of the incircle of $CBD$. Let $M$ be the center of the incircle of $ABD$. Let $L$ be the centroid of $DAC$. Let $N$ be the centroid of $BAC$.

Suppose $AK$, $BL$, $CM$, $DN$ have one common point.

Is $ABCD$ necessarily regular?

Answer 1

Nope, $AC$ need not equal to other $5$ edges and the tetrahedron need not be regular.

As long as the remaining $5$ edges are equal in length, we have

• $BC = CD = DB \implies \triangle BCD$ is equilateral $\implies K$ coincides with centroid of $\triangle BCD$.
• $AB = BD = AD \implies \triangle ABD$ is equilateral $\implies M$ coincides with centroid of $\triangle ABD$.

The $4$ lines $AK, BL, CM, DN$ will then intersect at the centroid of tetrahedron $ABCD$.