Geometry Problem. Post your solution in the comment box below.

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

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## Saturday, June 4, 2016

### Geometry Problem 1223: Circle, Parallel Chords

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Is arc AC=arcBD=arcEF. ThereforeCE//AF.

ReplyDeleteAPOSTOLIS MANOLOUDIS 4 HIGH SHCOOL OF KORRYDALLOS PIRAEUS GREECE

BDFE is a concyclic trapezoid, hence

ReplyDelete< EBD = < FEB = < ECD = @ say ....(1)

Similarly ABDC is a concyclic trapezoid, hence < ACD = < CDB = CEB = @ say ...(2)

From (1) and (2) < CEF = < ACE

But < ACE and < AFE are supplementary

Hence < CEF and < AFE are supplementary implying that AF//CE

Sumith Peiris

Moratuwa

Sri Lanka

Another method........

ReplyDeleteSince BDFE and ABDC are concyclic trapezoids AC = BD = EF

So in concyclic ACEF, AC = EF and the result follows

arc AC = arc BD, arc EF = arc BD

ReplyDelete=> arc EF = arc AC