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