Thales Theorem

Thales theorem states that if a triangle is inscribed in a circle, and one of the sides of the triangle is a diameter of the circle, then the angle opposite this side is a right angle.

Proof

To prove Thales theorem, we need to show that the opposite angle is a right angle. Let's consider a triangle ABC inscribed in a circle with center O and diameter BC.

Step 1: Draw a line from point A perpendicular to diameter BC. Let's call the point where the line intersects BC as D.

Step 2: Since OA is a radius of the circle, it is perpendicular to BC. Therefore, triangles OAD and OCD are right triangles.

Step 3: Since the angle at O is common to both triangles OAD and OCD, we can see that they are similar triangles.

Step 4: Since the triangles are similar, we can write the following proportion:

AD/CD = OA/OC

Step 5: Since OA = OC (both are radii of the circle), the above proportion can be simplified to:

AD/CD = 1

Step 6: Therefore, AD = CD, which means that triangle ABC is an isosceles triangle.

Step 7: Since triangle ABC is isosceles, the angles opposite the equal sides are equal. Therefore, angle ABD = angle ACD.

Step 8: Since angle ABD and angle ACD are angles in a straight line (180 degrees), their sum is 180 degrees.

Step 9: Therefore, angle ABD + angle ACD = 180 degrees.

Step 10: Substituting angle ACD with angle ABD (from step 7), we get:

angle ABD + angle ABD = 180 degrees

Step 11: Simplifying step 10, we get:

2 x angle ABD = 180 degrees

Step 12: Dividing both sides by 2, we get:

angle ABD = 90 degrees

Step 13: Therefore, the angle opposite the diameter BC is a right angle.

Conclusion

Hence, Thales theorem is proven.