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 Problem 39. Triangle, Incircle, Cyclic Quadrilaterals. Level: High School, SAT Prep, College



 

 

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:

  1. AGFO is a cyclic quadrilateral

  2. CHDO is a cyclic quadrilateral

  3. AGFE  is a cyclic quadrilateral

  4. CHDE  is a cyclic quadrilateral

  5. Angles FAG, FEG and FOG are congruent

  6. Angles DCH, DEH and DOH are congruent

  7. AG is perpendicular to CG and CH is perpendicular to AH

  8. AG is parallel to ED and CH is parallel  to EF

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

  10. Angles ABC and GEH are congruent

  11. Triangles ABC and GEH are similar

  12. AGHC is a cyclic quadrilateral

  13. Angles CAH, EGO, EFO, and CGH are congruent

  14. Angles ACG, EHO, EDO, and AHG are congruent

  15. 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.

 



 


 

 

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Last updated: July 1, 2007