1. **Problem Statement:** You want to understand different concepts step-by-step: dot, cross, box products; conversion between Cartesian and polar coordinates; and geometric properties of an equilateral triangle including centers and areas. 2. **Dot, Cross, and Box Products:** - The dot product for vectors $\vec{A}=(x_1,y_1)$ and $\vec{B}=(x_2,y_2)$ is: $$\vec{A} \cdot \vec{B} = x_1x_2 + y_1y_2$$ - The cross product in 2D (scalar value) is: $$\vec{A} \times \vec{B} = x_1y_2 - y_1x_2$$ - The box product (scalar triple product) typically involves three vectors in 3D; since given points are 2D, it may not apply here. 3. **Polar Coordinates Conversion:** Given a point $P(x,y)$: - Radius: $$r = \sqrt{x^2 + y^2}$$ - Angle: $$\cos \theta = \frac{x}{r}, \quad \sin \theta = \frac{y}{r}$$ From two points $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$ with polar coordinates $(r_1, \theta_1)$ and $(r_2, \theta_2)$ respectively, you can apply these individually. 4. **Key Triangle Centers and Properties:** - For triangle $ABC$ with vertices $A(6,4), B(-1,2), C(5,1)$: - Centroid $G$ is the average of coordinates: $$G = \left( \frac{6 - 1 + 5}{3}, \frac{4 + 2 + 1}{3} \right) = (\frac{10}{3}, \frac{7}{3})$$ - Orthocenter $H$ and Circumcenter $O$ can be found from intersections of altitudes and perpendicular bisectors respectively. 5. **Given Relationships:** - It is given (or to verify): $$HC = 2 \times GO$$ $$G - H = 2 (O - G)$$ Where $H$ = Orthocenter, $G$ = Centroid, $O$ = Circumcenter. 6. **Areas and Perimeters:** - Area $S_{ABC}$ and perimeter $P_{ABC}$ of triangle can be calculated with: - Area: $$S_{ABC} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$$ - Perimeter: $$P_{ABC} = |AB| + |BC| + |CA|$$ - Similarly, parameters $r$ (small gamma), $R$ (circumradius), $T$ (big capital Gamma), and the respective areas and perimeters of smaller segments or 'gammas' inside the triangle relate to concentric circles. 7. **Summary:** - Stepwise approach involves computing vector products, converting Cartesian to polar coordinates, computing centers of triangle and verifying centroid, orthocenter, circumcenter relations. - Use explicit formulas provided for area, perimeter, and coordinate conversions. - For Q3, implicit and explicit analytical equations of the circles and triangle sides come from the coordinate geometry of the points. This outline prepares you to tackle the problems systematically using the given definitions and relationships.