Problem 4

Let

A_0B_0C_0

and

A_1B_1C_1

be any two acute-angled triangles. Consider all triangles

ABC

similar to

A_1B_1C_1

(with corresponding vertices) and circumscribed about

A_0B_0C_0

, where

A_0

lies on

BC

B_0

CA

, and

C_0

AB

. Determine and construct the triangle of maximum area.

Step 1 of 7: Choose a reference triangle

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A_0\in YZ,\quad B_0\in ZX,\quad C_0\in XY

Detailed analysis

Choose any admissible triangle $XYZ$ similar to $A_1B_1C_1$ , labeled so that $A_0$ lies on $YZ$ , $B_0$ on $ZX$ , and $C_0$ on $XY$ . Such a triangle is obtained by choosing a line through $A_0$ and drawing the other two sides at the prescribed angles.