# A conic tangent to two triangles

conic sectionsprojective-geometry

I'm trying to prove the claim:

Two triangles inscribed in a conic section, there must be another conic tangent to the six edges of the two triangles.

The problem looks simpler than this question: A conic inside a hexagon , but I haven't found a way to use Pascal + Desargues + Brianchon directly.

Here is my approach:

1. Pascal's theorem tells us that ADBE lies on line GJ;
2. Pappus's theorem applying on ALE and BHD tells us that G, ADBE and LDHE are collinear, so LDHE lies on GJ;
3. Pappus's theorem applying on LKE and HID tells us that LIHK, LDHE and J are collinear, so LIHK lies on GJ, i.e. GJ, HK and LI are concurrent;
4. The converse of Brianchon's theorem tells us that there must be a conic tangent to the hexagon GHIJKL.

Are there any simple ways to prove it?

Your proof is already pretty short, but it can be shorter. For the hexagon $$\pmb{H}=AGBDJE$$ you've shown that the diagonals $$AD,GJ,BE$$ are concurrent. By the converse of Brianchon, $$\pmb{H}$$ has an inscribed conic. But the sides of $$\pmb{H}$$ are the same as the sides of the triangles, so we're done.