(b+c)/12=(c+a)/13=(a+b)/15.
13b+13c=12c+12a, c=12a-13b;
15b+15c=12a+12b, 3b=12a-15c, b=4a-5c;
15c+15a=13a+13b, 2a=13b-15c, a=½(13b-15c).
Substitute for a in c=12a-13b:
c=6(13b-15c)-13b=78b-13b-90c, 91c=65b, 7c=5b, b=7c/5.
Substitute for a in b=4a-5c:
b=2(13b-15c)-5c=26b-30c-5c=26b-35c, 25b=35c, 5b=7c, b=7c/5, confirming consistency.
a=½(13b-15c)=½(13(7c/5)-15c)=½(91c/5-15c)=(1/10)(91c-75c)=1.6c (8c/5).
CHECK: (b+c)/12=(7/5+1)c/12=(12/5)c/12=c/5 (=0.2c); (c+a)/13=2.6c/13=c/5; (a+b)/15=(1.6c+1.4c)/15=c/5, so:
a=8c/5, b=7c/5, meaning that we can work everything out in terms of c.
Cosine Rule:
a2=b2+c2-2bccosA; b2=a2+c2-2accosB; c2=a2+b2-2abcosC.
64c2/25=49c2/25+c2-14c2cosA/5, 64/25=49/25+1-14cosA/5, 14cosA/5=1+49/25-64/25=1-3/5=2/5, 14cosA=2, cosA=1/7.
49c2/25=64c2/25+c2-16c2cosB/5, 49/25=64/25+1-16cosB/5, 16cosB/5=1+64/25-49/25=8/5, 16cosB=8, cosB=1/2.
c2=64c2/25+49c2/25-112c2cosC/25, 1=113/25-112cosC/25, 112cosC/25=88/25, cosC=88/112=11/14.
cosA/2=cosB/7=cosC/11(=1/14) QED.