# Can you solve without trig?

Geometry Level 4

Triangle $$ABC$$ inscribed a circle $$(O;1)$$. If $$\angle A = 60 ^\circ$$ and $$\angle C = 45 ^\circ$$, 2 times the perimeter of the triangle can be written as $$a \sqrt{m} + b\sqrt{n} + c\sqrt{p}$$, where $$a,b,c,m,n,p$$ are positive integers and $$m,n,p$$ are distinct squarefrees. Find $$abc+mnp$$.

Note: 'Cause I don't like trigonometry, please post solutions using only trigonometric functions of these angles: $$30 ^\circ,45 ^\circ,60 ^\circ$$

Inspired by my math exercise.

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