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Proof of the Pythagorean Theorem

 

Statement of the Theorem

 

In a right triangle with legs a and b, and hypotenuse c, the Pythagorean Theorem states:

 

a² + b² = c²

 

Proof Using Similar Triangles (Euclid’s Proof)

 

Step 1: Construct the Right Triangle and Altitude

 

Consider a right triangle â–³ABC with a right angle at B. Draw the altitude BD from B to AC, splitting â–³ABC into two smaller right triangles: â–³ABD and â–³BDC.

 

Step 2: Observe the Similar Triangles

 

The two smaller triangles, â–³ABD and â–³BDC, are similar to â–³ABC by the AA similarity criterion, since:

• Each triangle has a right angle.

• Each smaller triangle shares another common angle with △ABC.

 

Thus, we conclude:

 

â–³ABD ~ â–³ABC and â–³BDC ~ â–³ABC

 

Step 3: Set Up Proportions Using Similarity

 

Since â–³ABD ~ â–³ABC, we can write the proportion:

 

AB / AC = BD / BC

 

Multiplying both sides by AC, we obtain:

 

AB² = AC × BD

 

Similarly, since â–³BDC ~ â–³ABC, we write:

 

BC / AC = BD / AB

 

Multiplying both sides by AC, we obtain:

 

BC² = AC × CD

 

Step 4: Sum the Two Equations

 

Adding the two equations together:

 

AB² + BC² = AC × BD + AC × CD

 

Since BD + CD = AC, we substitute:

 

AB² + BC² = AC × (BD + CD) = AC²

 

Thus, we conclude:

 

a² + b² = c²

 

Conclusion

 

This proof demonstrates that the sum of the squares of the two legs of a right triangle equals the square of the hypotenuse. Therefore, the Pythagorean Theorem holds for all right-angled triangles.

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