## Slicing A=100 AB=AC

### Slicing A=100 AB=AC

Fie . Pe prelungirea lui se ia punctul astfel incat . Sa se afle .

(Crux)
Quae nocent docent
Marius Stănean

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### Re: Slicing A=100 AB=AC

Solutia 1 :

Fie un punct pe mediatoarea segmentului astfel incat triunghiurile si sa fie congruente.( si in acelasi semiplan determinat de dreapta )
Atunci si de unde triunghiul este echilateral . In triunghiul isoscel avem de unde si Solutia 2 :

Fie un punct in plan astfel incat triunghiul este echilateral. ( si in acelasi semiplan determinat de dreapta )
Avem : si .
De aici triunghiurile si sunt congruente. de unde triunghiurile si sunt congruente Mesaje: 251
Membru din: Lun Aug 06, 2012 3:35 pm

### Re: Slicing A=100 AB=AC

Amazing solutions!

Another one:
Extend to so that , take on so that , see that is equilateral, hence , also and . From , with we infer . With , , consequently is the circumcenter of , therefore .

Yet another one, shorter:
Construct the equilateral and on either side of . Then is a kite and, with and , we get an isosceles trapezoid. By angle chase we infer . We also see that , with .

Best regards,
sunken rock
A blind man sees the details better. sunken rock

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### Re: Slicing A=100 AB=AC

Another nice idea:
Construct the equilateral triangle $\triangle ACO$, $O,B$ on the same side of $AC$, then $\triangle OAD\equiv\triangle ABC$ (s.a.s.), so $\angle AOD=100^\circ$ and $OD=OC$, therefore $O$ is the circumcenter of $\triangle ADC$, wherefrom $\angle ODC=10^\circ$ and consequently $\angle ADC=30^\circ$.

Best regards,
sunken rock
A blind man sees the details better. sunken rock

Mesaje: 645
Membru din: Joi Ian 06, 2011 2:49 pm
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### Re: Slicing A=100 AB=AC

... Or construct the isosceles triangle ADO with $\widehat{AOD}=100^\circ$, C and O on the same side of AB; as triangles ADO and ABC are congruent (s.a.s.) it follows that triangle AOC is equilateral wherefrom O is the circumcenter of triangle ADC and $2m(\widehat{ADC})=m(\widehat{AOC})=60^\circ$.

Best regards,
sunken rock
A blind man sees the details better. sunken rock

Mesaje: 645
Membru din: Joi Ian 06, 2011 2:49 pm
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