A
circle of center O
inscribed in a
triangle ABC
is tangent to sides BC, AC, and AB at
points D, E, and F,
respectively.
Lines AO and FD meet at H, and lines CO and DF
meet at G. Prove the following:

AGFO is a
cyclic quadrilateral

CHDO is a cyclic quadrilateral

AGFE is a cyclic quadrilateral

CHDE is a cyclic quadrilateral

Angles FAG, FEG and FOG are
congruent

Angles DCH, DEH and DOH are congruent

AG is perpendicular to CG
and
CH is perpendicular to AH

AG is
parallel to ED and CH is parallel to EF

Angles BAC and EGH are congruent and angles ACB and EHG are
congruent

Angles ABC and GEH are congruent

Triangles ABC and GEH are
similar

AGHC is a cyclic quadrilateral

Angles CAH, EGO, EFO, and CGH are congruent

Angles ACG, EHO, EDO, and AHG are congruent

O is the
incenter of triangle EGH
Hints: See
Problem 39 Geometry Help
Using the deductive method, we start
with a few true statements (the axioms) and use them to prove
dozens, hundreds, or thousands, of other statements (the
theorems). Geometry was organized by the Greek mathematician
Euclid, and his deductive method is still used by mathematicians
today.
