## Monday, September 21, 2009

### Problem 358. Isosceles triangle 80-80-20, Circle, Angles, Congruence

Proposed Problem
Click the figure below to see the complete problem 358 about Isosceles triangle 80-80-20, Circle, Angles, Congruence.

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Complete Problem 358
Level: High School, SAT Prep, College geometry

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2. http://ahmetelmas-geo-geo-antonio.blogspot.com/

3. Let AD cut the circle at P and EF at Q.

Easily Tr. AEC is 80-20-80 and so Tr. CEF is equilateral. Hence < DAF = 1/2 *60 - 20 = 10 and similarly < EFA = 1/2 *< ECA = 10. So AEPF is an isoceles trapezoid with AE = PF

Now since < ACF = 80, < PFD = 140 and since AD = BD, < PDF = DPF = 40. Hence PF = DF and so AE = DF

Sumith Peiris
Moratuwa
Sri Lanka

4. Problem 358
Triangle ECF is equilateral. The EF intersects AC in P (P is left of A ).Then <PEA=<EPA=40 so PA=AE.But triangle ACD= triangle FCP (AC=CF, <CPF=40=ADC,<PCF=<ACD=80). Therefore
CD=PC or CF+FD=CA+AP so DF=AP=AE.
APOSTOLIS MANOLOUDIS 4 HIGH SHCOOL OF KORYDALLOS PIRAEUS GREECE