Given:
- ABCD is an equilateral four-sided figure.
- Point O is taken inside ABCD such that its distance from the angular points D and B are equal.

To prove:
AO and OC are in the same straight line.

Proof:

Step 1: Draw Diagram

Let's start by drawing the equilateral four-sided figure ABCD. Inside this figure, we have point O.

Step 2: Analyze the Given Information

The given information tells us that the distance between point O and the angular points D and B are equal. Let's denote this distance as "x".

Step 3: Extend AO and OC

To prove that AO and OC are in the same straight line, we need to extend AO and OC until they meet. Let's denote the point of intersection as point P.

Step 4: Triangle AOD and Triangle BOD

Since ABCD is an equilateral four-sided figure, all sides are equal in length. Let's denote the length of each side as "a".

In triangle AOD:
- Side AO = x (Given)
- Side OD = a (Side of the equilateral figure ABCD)
- Side AD = a (Side of the equilateral figure ABCD)

In triangle BOD:
- Side BO = x (Given)
- Side OD = a (Side of the equilateral figure ABCD)
- Side BD = a (Side of the equilateral figure ABCD)

Step 5: Congruent Triangles

We can observe that triangle AOD and triangle BOD have two sides of equal length. Additionally, they share the side OD, which is also of equal length.

Using the Side-Side-Side (SSS) congruence criterion, we can conclude that triangle AOD and triangle BOD are congruent.

Step 6: Angle AOD and Angle BOD

Since triangle AOD and triangle BOD are congruent, their corresponding angles are also equal.

Angle AOD = Angle BOD

Step 7: Angle AOC

Now, let's consider triangle AOC.

Angle AOC = Angle AOD + Angle BOD (By Angle Addition Property)

Angle AOC = Angle AOD + Angle AOD (Since Angle AOD = Angle BOD)

Angle AOC = 2 * Angle AOD

Step 8: Angle AOC and Angle OAC

In triangle AOC, let's consider angle OAC.

Angle OAC is an exterior angle to triangle AOD.

Angle OAC = Angle AOD + Angle BOD (By Exterior Angle Theorem)

Angle OAC = Angle AOD + Angle AOD (Since Angle AOD = Angle BOD)

Angle OAC = 2 * Angle AOD

Step 9: Angle AOC and Angle OAC

From steps 7 and 8, we can conclude that Angle AOC = Angle OAC.

Step 10: AO and OC are in the Same Straight Line

Since Angle AOC = Angle OAC, we can conclude that AO and OC are in the same straight line.

Therefore, we have proved that AO and OC are in the same straight line.