A triangle is drawn by joining a point on a semicircle to the end of the diameter. Then semicircles are drawn with the other two sides as diameter.


Prove that the sum of the areas of the blue and red crescents in the second picture is equal to the area of the triangle.

From, the given diagram,


∠ACB = 90° (since, angle on a semicircle is right angle)


Thus, by Pythagoras theorem


AC2 + CB2 = AB2


⇒ a2 = b2 + c2 …..(1)


Also,




Also,
where, r = radius


Here,



Now, by second diagram



We get,


where, r = radius


Here,



where, r = radius


Here,



Thus,


Sum of Area of crescent = S2 + S3 + area of Δ ABC – S1




(from(1))


⇒ Sum of Area of crescent = area of Δ ABC


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