
Re: geometry puzzle
Posted:
Oct 8, 2013 10:45 PM


> 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.

