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Geometry ProblemLevel: Mathematics Education, High School, Honors Geometry, College.Click the figure below to see the complete problem 634.
Note that Triangles ABC, BHC and ABH are similar triangles ( Case AA)In each triangle , ratio of height from 90 degrees angle/ radius of incenter circle = h1/r1= h2/r2=h3/r3So ratio S(AEB)/S(AHB)=S(BFC)/S(BHC)=S(ADC)/S(ABC)But we have S(ABC)=S(ABH)+S(BHC)So S(AEB)+S(BFC)=S(ABC)Peter Tran
Call α = angle A, AB = c, BC = a and AC = b.Because AE, BF and CD are bisectors, it is easy to see that ∆AEB, ∆BFC and ∆ACD all have angles ½α, 45º-½α and 135º, so they are all similar triangles.factor ∆ACD → ∆ABE = c/b so S1 = S3*(c/b)².factor ∆ACD → ∆BCF = a/b so S2 = S3*(a/b)².S1 + S2 = S3*((c/b)² + (a/b)²) = S3*(a²+c²)/b²Pythagoras in ∆ABC: a²+c²=b²S1 + S2 = S3*b²/b² = S3.QED.(Note: there is a little error in the text on the side of the picture. F is the incenter of ∆BHC.)
Thanks Henkie for the note.
All 3 coloured triangles are similar having angles A/2, 45-A/2 and 135Hence S1/c^2 = S2/a^2 = S3/b^2 = (S1+S2)/(c^2+a^2) = (S1+S2)/b^2 using Pythagoras So S1 + S2 = S3Here I've used the well known algebraic tool that if x/p = y/q then each ratio = (x+y)/(p+q)Sumith PeirisMoratuwaSri Lanka
Leave it to fellow Gogeometers to prove that AEFC is con cyclic
Also that D is the orthocentre of Tr. BEF