To solve this problem, let's break it down step by step using the properties of segment addition and subtraction.

Given Information:

1. Points A, C, B, D are on a straight line in the order A, C, B, D.
2. $AB = CD$
3. $AC = BD$
4. Point I is the midpoint of BC and the midpoint of AD.
5. Line d is perpendicular to xy at point C, and point M is an arbitrary point on line d.

Steps to Solve:

1. Set Up the Points:

   • Place the points on a coordinate system.
   • Let $A(0, 0)$, $C(a, 0)$, $B(a + b, 0)$, and $D(a + b + c, 0)$, where $b$ is the length of segment AB and $c$ is the length of segment CD.
   • Given $AB = CD$, $b = c$.

2. Find Coordinates for Points:

   • Since $I$ is the midpoint of both segments:
     • $I$ for BC: $I = \left(\frac{a + (a + b)}{2}, 0\right) = \left(a + \frac{b}{2}, 0\right)$
     • $I$ for AD: $I = \left(\frac{0 + (a + b + c)}{2}, 0\right) = \left(\frac{a + b + c}{2}, 0\right)$
     • Set $b = c$ implies that $a + \frac{b}{2} = \frac{a + 2b}{2}$.

3. Draw the Perpendicular Line:

   • At point $C(a, 0)$, draw line $d$ which is vertical, meaning $M$ can be any point of the form $(a, y)$ where $y$ is not equal to 0.

4. Show that $MA = MB$:

   • The distances from $M$ to $A$ and $B$ can be calculated:
     • Distance $MA$: $MA = \sqrt{(a - 0)^2 + (y - 0)^2} = \sqrt{a^2 + y^2}$
     • Distance $MB$: $MB = \sqrt{(a - (a + b))^2 + (y - 0)^2} = \sqrt{(-b)^2 + y^2} = \sqrt{b^2 + y^2}$
   • Since $AB = CD$ (or $b = c$), and with $b = c$, $MA = MB$.

Conclusion:

Since both distances are equal for any arbitrary point $M$ on line $d$, we conclude that $MA = MB$.

Graph:

To visualize this, you can draw:

• A horizontal line with points A, C, B, and D marked at appropriate distances.
• A vertical line at point C.
• Points A, B, and M marked clearly.

Justification:

The geometric properties of midpoints and the equal distances from a perpendicular point to two points on a straight line show that the distances $MA$ and $MB$ remain equal.

Questions for Further Understanding:

1. How would the distances change if $b \neq c$?
2. What happens if point M lies on the x-axis instead of line d?
3. Can you identify similar properties in other geometric figures?
4. How would you prove the midpoint theorem using different shapes?
5. What role does symmetry play in this problem?

Tip:

When solving geometric problems, always visualize them with a sketch to better understand relationships and properties.