Three triangles are formed by drawing lines from the vertices of a triangle $ABC$ to the opposite sides, each making equal angles with the sides. Let $\Delta_1, \Delta_2, \Delta_3$ be the areas of the three smaller triangles formed at the vertices with the circumradius equal to 1. a) Express the total area $\Delta = \Delta_1 + \Delta_2 + \Delta_3$ in terms of $A, B, C$. b) Find the angles $A, B, C$ that maximize $\Delta$. c) Verify that the maximum occurs for an isosceles triangle and prove $C = A$ by calculus.
a) $\Delta_1 = \tfrac{1}{2}\cdot 1\cdot 1\cdot\sin 2A$, $\Delta_2 = \tfrac{1}{2}\cdot 1\cdot 1\cdot\sin 2B$, $\Delta_3 = \tfrac{1}{2}\cdot 1\cdot 1\cdot\sin 2C$, $\Delta = \tfrac{1}{2}(\sin 2A + \sin 2B + \sin 2C)$. b) $A+B+C=\pi \Rightarrow \Delta = \tfrac{1}{2}(\sin 2A + \sin 2B - \sin 2(A+B))$. $\frac{d\Delta}{dA} = \cos 2A - \cos 2(A+B) = 0 \Rightarrow \cos 2A = \cos 2(A+B) \Rightarrow 2A = 2\pi - (2A+2B) \Rightarrow 2A+B=\pi$. Since $A+B+C=\pi$, we get $A=C$. c) Consider isosceles and then prove $C=A$ by calculus.
Three triangles are formed by drawing lines from the vertices of a triangle $ABC$ to the opposite sides, each making equal angles with the sides. Let $\Delta_1, \Delta_2, \Delta_3$ be the areas of the three smaller triangles formed at the vertices with the circumradius equal to 1.
a) Express the total area $\Delta = \Delta_1 + \Delta_2 + \Delta_3$ in terms of $A, B, C$.
b) Find the angles $A, B, C$ that maximize $\Delta$.
c) Verify that the maximum occurs for an isosceles triangle and prove $C = A$ by calculus.