See complete Problem 179 at:
www.gogeometry.com/problem/p179_rhombus_square_diagonal_angle.htm
Rhombus, Square, Diagonal, Angle. Level: High School, SAT Prep, College geometry
Post your solutions or ideas in the comments.
Monday, September 15, 2008
Elearn Geometry Problem 179
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DA=DC=DE, D is the circumcenter of CAE, mAEC=90°/2=45°
ReplyDeletenotice that AD = CD = DE so if we draw a circle with centre D and radius DE, say, then it will pass through A, C and E. but angle ADC is 90 degrees and it is also the central angle. so angle AEC will be half of angle ADC that is angle AEC = 45 degrees.
ReplyDeleteQ.E.D.
D is the Centre of circle ACE and the result follows
ReplyDeleteSee the drawing : Drawing
ReplyDeleteCDEF Rombus =>DC=DE
ABCD square =>DC=DA
DA=DC=DE => A, C and E are on the circle with center D and radius AD
With the theorem of inscribed angle, the central angle of a circle is twice any inscribed angle => ∠ ADC=2 ∠ CEA
∠ ACD=90° => ∠ CEA=45°
Let <CFE=<CDE=2x
ReplyDelete<ADE=90+2x
AD=AE, <DEA=45-x
FE=FC, <CEF=90-x
<DEF=180-<CDE=180-2x
Since <DEF=<DEA+<AEC+<CEF
180-2x=45-x+<AEC+90-x
<AEC=45