Proof:

Given: Quadrilateral ABCD, BN and DM are drawn perpendicular to AC such that BN = DM.

To prove: O is the midpoint of BD.

Proof:

Step 1: Drawing a diagram

First, let's draw a diagram to better understand the given information.

[insert diagram here]

Step 2: Analyzing the given information

From the given information, we know that BN = DM. This implies that the lengths BN and DM are equal.

Step 3: Defining the midpoint

We need to prove that O is the midpoint of BD. To do that, we first need to define what a midpoint is.

A midpoint is a point that divides a line segment into two equal parts. In this case, we need to show that O divides BD into two equal parts.

Step 4: Proving O is the midpoint of BD

To prove that O is the midpoint of BD, we need to show that OB = OD.

Step 5: Using congruent triangles

Let's consider triangles BNO and DMO.

From the given information, we know that BN = DM. Additionally, both triangles have a right angle (90 degrees) as BN and DM are perpendicular to AC.

Therefore, by the hypotenuse-leg congruence theorem, we can conclude that triangle BNO is congruent to triangle DMO.

Step 6: Proving OB = OD

Since triangle BNO is congruent to triangle DMO, their corresponding sides are also congruent.

Therefore, OB = OD.

Step 7: Concluding O is the midpoint of BD

Since OB = OD, we can conclude that O is the midpoint of BD.

Conclusion:

Therefore, we have proved that O is the midpoint of BD in quadrilateral ABCD, where BN and DM are drawn perpendicular to AC with BN = DM.