New Year's Countdown Day 6: Hexagon in Triangle

Geometry Level 3

Triangles A B C ABC and D E F DEF share the same incircle. The intersection of the two triangles forms a hexagon. Are the three long diagonals of this hexagon concurrent i.e. do they intersect at only one point?

This problem is part of the set New Year's Countdown 2017 .
Yes, but only for a finite number of choices for △ A B C , D E F \triangle ABC, DEF Yes, for any △ A B C , D E F \triangle ABC, DEF No, never Yes, but only if △ A B C ∼ △ D E F \triangle ABC \sim \triangle DEF Yes, but only if △ A B C ≅ △ D E F \triangle ABC \cong \triangle DEF

Steven Yuan by Steven Yuan , 20, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Steven Yuan
Dec 7, 2017

The hexagon has an inscribed circle. Thus, by Brianchon's Theorem , its long diagonals are concurrent. This is true for any choice of triangles A B C , D E F , ABC, DEF, as long as they share the same incircle. The answer is yes, for any △ A B C , D E F \triangle ABC, DEF .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...