Proof:

Given:
- AD is perpendicular to BC
- BD/AD = DA/DC

To prove:
Triangle ABC is a right triangle

Proof by contradiction:

Assume that triangle ABC is not a right triangle. This means that the angle at vertex B is not a right angle.

Property of Perpendicular Lines:
- When a line is perpendicular to another line, it creates four right angles.

Property of Similar Triangles:
- If two triangles have the same angles, then they are similar.

Construction:
Construct a line segment CE perpendicular to AB at point E.

Proof:
Since AD is perpendicular to BC, it forms a right angle at D. Therefore, triangle ABD is a right triangle.

Using the given proportion BD/AD = DA/DC, we can rewrite it as BD/DA = AD/DC.

Using the property of similar triangles, we can conclude that triangle ABD is similar to triangle ADC.

Since triangle ABD is a right triangle, angle ADB is a right angle.

Since triangle ABD is similar to triangle ADC, angle ADC is also a right angle.

Since angle ADC is a right angle, line segment CE is parallel to line segment AD.

Since line segment CE is parallel to line segment AD and line segment AD is perpendicular to line segment BC, it follows that line segment CE is perpendicular to line segment BC.

Since line segment CE is perpendicular to line segment BC, it forms a right angle at C. Therefore, triangle ABC is a right triangle.

Conclusion:
Thus, we have proved that if AD is perpendicular to BC and BD/AD = DA/DC, then triangle ABC is a right triangle.