1. Zadatak

Neka je $x$ duljina stranice trokuta, a $\varphi$ kut $\angle BAE$.

Površine trokuta su: $$\begin{aligned} P_{\triangle ABE}&=\frac{x\cos\varphi\cdot x\sin\varphi}{2}=\frac{x^2}4\sin2\varphi\\ P_{\triangle AFD}&=\frac{x\cos(30^\circ-\varphi)\cdot x\sin(30^\circ-\varphi)}{2}=\frac{x^2}4\sin(60^\circ-2\varphi)\\ P_{\triangle ECF}&=\frac{x\cos(60^\circ-\varphi)\cdot x\sin(60^\circ-\varphi)}{2}=\frac{x^2}4\sin(120^\circ-2\varphi) \end{aligned}$$ Adicijski teoremi: $$\begin{aligned} P_{\triangle ECF}&=\frac{x^2}4(\sin120^\circ\cos2\varphi-\cos120^\circ\sin2\varphi)\\ &=\frac{x^2}4\left(\frac{\sqrt3}2\cos2\varphi+\frac12\sin2\varphi\right)\\ P_{\triangle ABE}+P_{\triangle AFD}&=\frac{x^2}4(\sin60^\circ\cos2\varphi-\cos60^\circ\sin2\varphi+\sin2\varphi)\\ &=\frac{x^2}4\left(\frac{\sqrt3}2\cos2\varphi+\frac12\sin2\varphi\right)=P_{\triangle ECF} \end{aligned}$$