Gogeometry Problem 975


From Gogeometry

Solution

Solution

Using :

Additional points

Similar triangles

Solution
 •ΔABC is isosceles and right in B => BA=BC and ∠BAC=∠BCA=Π/4
 • Define G in AC such as AB=AG => ΔBAG is isosceles in A
 •Define H intersection of AD and BG
 •∠BAH=∠GAH=Π/4/2=Π/8  and HB=HG and ∠AHB=∠AHG=Π/2 
 •and ∠ABH=∠AGH and ∠ABH+ ∠BAH =∠AGH + ∠GAH =Π/2
 •∠ABD= ∠ABH+ ∠HBD =Π/2 => ∠HBD= ∠BAH
 •=> ∠AGD= Π/2
 •In ΔDGC, ∠AGD= Π/2 and ∠GCD= Π/4 => ∠GDC= Π/4 
 •=> ΔDGC is isosceles => GD=GC
 •But GD=BD => GC=BD
 •AC=AG+GC
 •Therefore AC=AB+BD 

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