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Topic: geometry puzzle
Replies: 6   Last Post: Nov 12, 2013 4:55 AM

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Peter Scales

Posts: 192
From: Australia
Registered: 4/3/05
Re: geometry puzzle
Posted: Oct 8, 2013 10:45 PM
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> Given a triangle, with sides A, B, C, and
> opposite angles a, b, c.
> Prove: if A < (B + C)/2, then a < (b + c)/2

Hi Rich,
Start with an equilateral triangle, base A1
Clearly A1=(B1+C1)/2 and a1=(b1+c1)/2

Now increase the height to give an isosceles triangle with base A2
clearly a2<60deg, so b2 and c2 are both >60
: a2<(b2+c2)/2 and A2<(B2+C2)/2

Now translate the apex sideways to give a scalene triangle in which a3<a2
: b3+c3>b2+c2
: a3<(b3+c3)/2
But A3=A2 and B3+C3>B2+C2
: A3<(B3+C3)/2 QED

Regards, Peter Scales.

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