1. **Problem Statement:** We have an oval (ellipse) with radius $R=74$ and a square inside it with side length $S=46$. Points $a$, $b$, and $c$ are on these shapes, and we want formulas to find distances $ab$ and $ac$ (noting $ab \neq bc$). 2. **Understanding the Shapes:** - The oval is likely an ellipse with semi-major or semi-minor axis $R=74$. - The square inside has side $S=46$. - Distances given: vertical height of oval $=102.459$, vertical distances from center to square edges $=65.050$ (up) and $41.200$ (down). 3. **Coordinate Setup:** Assume the oval is centered at origin $(0,0)$. - Let the square be positioned such that its center aligns vertically with the oval's center. - Points $a$, $b$, and $c$ lie on or inside these shapes. 4. **Formulas for Distances:** - Distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 5. **Finding Coordinates of Points:** - If $a$, $b$, and $c$ coordinates are known or can be expressed in terms of $R$ and $S$, plug into the distance formula. 6. **Example:** - Suppose $a=(x_a,y_a)$ on oval boundary. - Suppose $b=(x_b,y_b)$ on square boundary. - Then: $$ab = \sqrt{(x_b - x_a)^2 + (y_b - y_a)^2}$$ - Similarly for $ac$. 7. **Ellipse Equation:** If oval is ellipse with semi-major axis $a=R=74$ and semi-minor axis $b$, ellipse equation: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ 8. **Square Coordinates:** Square side $S=46$, so half side $=23$. If centered at $(0,y_c)$, corners are at: $$(\pm 23, y_c \pm 23)$$ 9. **Summary:** - To find $ab$ and $ac$, determine coordinates of $a$, $b$, and $c$. - Use distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ **Final formulas:** $$ab = \sqrt{(x_b - x_a)^2 + (y_b - y_a)^2}$$ $$ac = \sqrt{(x_c - x_a)^2 + (y_c - y_a)^2}$$ These formulas allow calculation of distances once coordinates are known or expressed in terms of $R$ and $S$.