MHB Find the Special Point of a Triangular Pyramid: Proof Explained

[veronica1999]

Suppose that the lateral faces VAB, VBC, and V CA of triangular pyramid VABC
all have the same height drawn from V . Let F be the point in plane ABC that is closest
to V , so that VF is the altitude of the pyramid. Show that F is one of the special points
of triangle ABC.

I made the triangular pyramid and I think the special point is the median.
Am I correct?

Thanks.

 

[Opalg]

Suppose that the lateral faces VAB, VBC, and VCA of triangular pyramid VABC
all have the same height drawn from V . Let F be the point in plane ABC that is closest
to V , so that VF is the altitude of the pyramid. Show that F is one of the special points
of triangle ABC.

I made the triangular pyramid and I think the special point is the median.
Am I correct?

Thanks.

Let $d$ be the "height drawn from $V$" of the three lateral faces. The sphere of radius $d$ centred at $V$ touches (tangentially) each of the three sides $BC$, $CA$, $AB$, of the base of the pyramid. Therefore the intersection of the sphere with the plane $ABC$ is the incircle of the triangle $ABC$. The line $VF$ is perpendicular to the plane $ABC$, so that $F$ is the centre of that circle. So I reckon that $F$ is the incentre of the triangle.