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

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

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## Friday, November 4, 2016

### Geometry Problem 1283 Two Equilateral Triangle, Perpendicular, Midpoint

Labels:
equilateral,
midpoint,
perpendicular,
vertex

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https://goo.gl/photos/96rSC1H87PN4Xb7g7

ReplyDeleteLet N is the projection of M over CDE

In triangle EMD, MN is the median and altitude from M => MED is isosceles triangle

Connect CM , CM ⊥AB and BE⊥EC => quadri. CMBE is cyclic

in cyclic quadrilateral CMBE ∠MDC=∠MBC=60 degrees => Triangle EMC is equilateral

Problem 1283

ReplyDeleteSuppose that MN is perpendicular to CE ( DN =NE ,ABED is trapezoid ) so triangle MED is isosceles (ME=MD).But <AMC=90=<ADC then AMDC is cyclic. So <MDA=<MCA=60/2=30.

Then <MDE=90-30=60.Therefore triangle MDE is equilateral.

APOSTOLIS MANOLOUDIS 4 HIGH SCHOOL KORYDALLOS PIRAEUS GREECE

BECM is cyclic with BC as diameter so

ReplyDelete< MEC = < MBC = 60

ACDM is cyclic with AC as diameter so

< MDE < MAC = 60

So 2 angles of Tr. MDE = 60 and is hence equilateral

Sumith Peiris

Moratuwa

Sri Lanka