Sunday Afternoon Maths XXXVIII

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$$(a^2+b^2)(c^2+d^2)$$ $$a^2c^2+b^2c^2+a^2d^2+b^2d^2$$ $$(bd-ac)^2=b^2d^2+a^2c^2-2abcd$$ $$(bc+ad)^2=b^2c^2+a^2d^2+2abcd$$ $$(a^2+b^2)(c^2+d^2)=(bd-ac)^2+(bc+ad)^2$$

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$$\frac{CD}{\sin(CBD)}=\frac{BD}{\sin(BCD)}$$ $$\frac{AD}{\sin(ABD)}=\frac{BD}{\sin(BAD)}$$ $$\sin(BAD)=\frac{BD\sin(ABD)}{AD}\\ =\frac{BD\sin(CBD)}{CD}\\ =\sin(BCD)$$ $$\frac{AB}{\sin(BCD)}=\frac{BC}{\sin(BAD)}$$ $$AB=BC$$

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$$\arctan(a)+\arctan(b)+\arctan(c)=\pi$$