How to Prove the Obtuse Rule, Book II Prop. 12 of Elements

Step-by-Step Guide

1. 1

   Step 1: Take a look at the diagram at hand.

   Triangle ABC is an obtuse angled triangle having the obtuse angle at C.
   
   Squares are constructed on each of the three sides AB, BC and AC.
   
   Further, BC is extended and from A, a perpendicular is dropped on the extended portion of BC which meets it at D.
2. 2

   Step 2: Understand the objective.

   We need to prove that the square constructed on the side which subtends the obtuse angle (the side AB) is larger than the other two squares taken together by twice the rectangle contained by either of the other two sides (say BC) and its extension (CD, in this case).
   
   This can be done if we prove that the area of the square constructed on side AB equals the sum of the areas of the squares on the other two sides plus twice the area of the rectangle which can be formed by BC and CD.
   
   Mathematically, we need to prove that AB2 = BC2 + AC2 +
   2.BC.CD ,,,,, Thus we have proved what we needed to. , 13 of Elements for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation.
   
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3. 3

   Step 3: Apply Pythagorean Theorem to triangle ABD to get: AB2 = AD2 + BD2 ...(1)

4. 4

   Step 4: We already know that: BD = BC + CD Squaring both sides

5. 5

   Step 5: BD2 = (BC + CD)2 => BD2 = BC2 + CD2 + 2.BC.CD ...(2)

6. 6

   Step 6: Substitute BD2 from (2) into (1) to get: AB2 = AD2 + BC2 + CD2 + 2.BC.CD ...(3)

7. 7

   Step 7: Apply Pythagorean Theorem in triangle ACD to get: AC2 = AD2 + CD2 ...(4)

8. 8

   Step 8: Substitute AD2 + CD2 = AC2

9. 9

   Step 9: from (4) into (3) to get: AB2 = BC2 + AC2 + 2.BC.CD.

10. 10

    Step 10: Make use of helper articles when proceeding through this tutorial: See the article How to Prove the Acute Rule

11. 11

    Step 11: Book II Prop.

Detailed Guide

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