1. **Problem statement:** We have a square with side length $S=46$ and an oval (ellipse) with radius $R=74$ intersecting the square vertically. Points $a$, $b$, and $c$ lie on the intersection of the oval and square boundaries. We want formulas to find the distances $ab$ and $bc$, noting $ab \neq bc$. 2. **Model the square:** The square is centered at the origin with side $S=46$, so its vertices are at $\left(\pm \frac{S}{2}, \pm \frac{S}{2}\right) = (\pm 23, \pm 23)$. 3. **Model the oval (ellipse):** Assuming the oval is a vertical ellipse centered at the origin with vertical radius $R=74$ and horizontal radius $a$ unknown. The ellipse equation is: $$\frac{x^2}{a^2} + \frac{y^2}{R^2} = 1$$ 4. **Find $a$ (horizontal radius):** Using the given horizontal distance $65.050$ (likely the horizontal distance from center to intersection on the right side), we approximate $a = 65.050$. 5. **Find intersection points:** The square boundary lines are at $x=\pm 23$ and $y=\pm 23$. Intersections occur where ellipse and square boundaries meet. - For vertical sides $x=\pm 23$, solve ellipse for $y$: $$y = \pm R \sqrt{1 - \frac{x^2}{a^2}} = \pm 74 \sqrt{1 - \frac{23^2}{65.050^2}}$$ Calculate: $$y = \pm 74 \sqrt{1 - \frac{529}{4231.5}} = \pm 74 \sqrt{1 - 0.125} = \pm 74 \times 0.935 = \pm 69.19$$ - For horizontal sides $y=\pm 23$, solve ellipse for $x$: $$x = \pm a \sqrt{1 - \frac{y^2}{R^2}} = \pm 65.050 \sqrt{1 - \frac{23^2}{74^2}}$$ Calculate: $$x = \pm 65.050 \sqrt{1 - \frac{529}{5476}} = \pm 65.050 \sqrt{1 - 0.0965} = \pm 65.050 \times 0.951 = \pm 61.87$$ 6. **Identify points $a$, $b$, and $c$:** - $a$ and $c$ are likely on the horizontal sides at $(\pm 61.87, \pm 23)$ - $b$ is on the vertical side at $(\pm 23, \pm 69.19)$ 7. **Calculate distances:** - Distance $ab$ between points $a=(61.87, 23)$ and $b=(23, 69.19)$: $$ab = \sqrt{(61.87 - 23)^2 + (23 - 69.19)^2} = \sqrt{38.87^2 + (-46.19)^2} = \sqrt{1511.5 + 2134.0} = \sqrt{3645.5} = 60.37$$ - Distance $bc$ between points $b=(23, 69.19)$ and $c=(-61.87, 23)$: $$bc = \sqrt{(23 + 61.87)^2 + (69.19 - 23)^2} = \sqrt{84.87^2 + 46.19^2} = \sqrt{7202.5 + 2134.0} = \sqrt{9336.5} = 96.62$$ **Final formulas:** $$ab = \sqrt{\left(a \sqrt{1 - \frac{S^2}{4a^2}} - \frac{S}{2}\right)^2 + \left(\frac{S}{2} - R \sqrt{1 - \frac{S^2}{4a^2}}\right)^2}$$ $$bc = \sqrt{\left(\frac{S}{2} + a \sqrt{1 - \frac{S^2}{4R^2}}\right)^2 + \left(R \sqrt{1 - \frac{S^2}{4a^2}} - \frac{S}{2}\right)^2}$$ where $S=46$, $R=74$, and $a$ is the horizontal radius of the oval (approx. 65.050). These formulas allow calculation of distances $ab$ and $bc$ for any given $a$, $R$, and $S$.