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Geometry ProblemLevel: Mathematics Education, High School, Honors Geometry, College.Click the figure below to see the problem 723 details.
First we observe that diagonal AOC is a diameter of circle (O) since <ADC is a rt angle.<AHC = <ADC = 90 degSimilarly <EHG = < EFG = 90 degBy Problem 722, A, E, H are collinear and so <AHG is the same as <EHG = 90 degHence <AHC + <AHG = 180 deg andC, H, G are collinear.
Solution 2:<DHG = <DEG = 45 deg<DHC = supplement of <DBC = 180deg - 45deg Sum = 180 degSo G, H, C are collinear.[Note: DB passes thro'O and bisects the rt <ABC]
We can prove this problem using result of problem 722 orNote that DF and DB are diameters of circles O and O’So (BHD)=(DHF)=90 >> B,H, F are collinear(GHF)=(CHB)=45 ----- (angles face 90 arc)So (CHG)=(BHF)+45-45= 180 >> C, H, G are collinear
Reference 722 < AHC = 90 = < EHG. Hence CHG are collinear