(Too) many circles

Geometry Level 3

Four semicircles and two circles are inscribed in a quadrangle P Q R S PQRS as shown.

  • P Q = 6 PQ=6
  • The radius of the small semicircles is 1, their midpoints are on P Q PQ
  • The radius of the large semicircle is 3, the midpoint is on R S RS
  • The radius of the two circles is 1.5

The length of P S PS is a b + c d a\sqrt{b}+c\sqrt{d} , where a , b , c , d a,b,c,d are integers and b , d b,d are square-free. Find a + b + c + d a+b+c+d .


The answer is 12.

Dick van der Leeden by Dick van der Leeden , 66, Netherlands

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1 solution

A B = 1 = A P , B C = 1.5 = C D , D E = 3 = E S AB=1=AP, BC=1.5=CD,DE=3=ES

P A = H F = 1 , C D = 1.5 ⇒ F C = 0.5 PA=HF=1,CD=1.5 \Rightarrow FC=0.5

H C = S G = 1.5 , E = 3 ⇒ G E = 1.5 HC=SG=1.5,E=3 \Rightarrow GE=1.5

A C = 1 + 1.5 = 2.5 AC=1+1.5=2.5

C E = 3 + 1.5 = 4.5 CE=3+1.5=4.5

A F 2 = 2. 5 2 − 0. 5 2 = 6 ⇒ A F = 6 AF^2=2.5^2-0.5^2=6 \Rightarrow AF = \sqrt{6}

C G 2 = 4. 5 2 − 1. 5 2 = 18 ⇒ C G = 3 2 CG^2=4.5^2-1.5^2=18 \Rightarrow CG = 3\sqrt{2}

P S = P H + H S = A F + C G = 6 + 3 2 PS=PH+HS=AF+CG=\sqrt{6}+3\sqrt{2}

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