Geometry Problem

Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the complete problem 980.

## Thursday, February 13, 2014

### Geometry Problem 980. Equilateral Triangle, Vertices, Three Parallel, Equal Circles, Construction

Labels:
circle,
construction,
equilateral,
parallel,
triangle,
vertex

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Because AC//DF and BD//CE, 60=<BOC=<BDF, 60=<AOQ=180-<ACF=<DFC, then <DFC=<BDF so OCDF is cyclic isosceles trapezoid. But 60=180-<QOD=<ECF so <DFC=<ECF which makes FEDC isosceles trapezoid as well. That means DEOCF is cyclic and OEF equilateral from subtended angles.

ReplyDeleteFrom simple angle chasing, < COD = 120 and < DEC = 60 so CODE is a isosceles trapezoid with CO = DO.

ReplyDeleteSimilarly CF = OD (from isosceles trapezoid CODF)

So Tr.s COF and ODE are congruent SAS, the included angle being 120

Hence OE = OF and < DOE + < COF = < DOE + < DEO = 60

So < FOE = 60 and the result follows

Sumith Peiris

Moratuwa

Sri Lanka