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Two Crescents (Posted on 2004-08-11) Difficulty: 2 of 5
Given a right triangle, draw the three following semicircles:
  1. The semicircle with diameter formed by one of the legs and extending away from the triangle.
  2. The semicircle with diameter formed by the other leg and extending away from the triangle.
  3. The semicircle with diameter formed by the hypotenuse and extending towards the triangle.

Prove that the area of the two crescents (shown in RED and BLUE) formed by the three semicircles equals the area of the triangle.

No Solution Yet Submitted by ThoughtProvoker    
Rating: 3.2000 (10 votes)

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Simple Solution | Comment 12 of 15 |

Let a=vertical triangle side, b=horizonta triangle side, and c=hypotenuse; semi-a=red semi-circle, etc.

Area(crescents) = Area(total) - A(semi-c)


Area(crescents)=(a) + (b) + Area(triangle) - (c)

Area(crescents) = Area(triangle) + ((a+b-c))

Since c=a+b, the third term is zero. Therefore,

Area(crescents) = Area(triangle)

  Posted by Tim Norwood on 2006-01-22 13:52:25
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