Circumcircles

Geometry Level 3

Consider △ A B C \triangle ABC with circumcenter O O and point M M the reflection of point O O across line A C AC . Line A M AM and line B C BC are extended to meet at point D D . If r 1 r_1 , r 2 r_2 , and r 3 r_3 are the circumradii of △ A B C \triangle ABC , △ A B D \triangle ABD , and △ A C D \triangle ACD respectively, what is r 1 : r 2 : r 3 r_1 : r_2 : r_3 ?

S i n B : C o s A : S i n C SinB : CosA : SinC S i n A : S i n C : C o s B SinA : SinC : CosB C o s A : S i n C : S i n B CosA : SinC : SinB S i n C : C o s B : S i n A SinC : CosB : SinA S i n B : S i n A : C o s C SinB : SinA : CosC C o s C : S i n B : S i n A CosC : SinB : SinA

Sahil Bansal by Sahil Bansal , 23, India

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1 solution

It is easy to get ∠ A D B = 90 ° − A \angle {ADB}=90\degree -A . So A C ‾ = 2 r 3 cos ⁡ A \overline {AC}=2r_3\cos {A} . Also, A C ‾ = 2 r 1 sin ⁡ B \overline {AC}=2r_1\sin { B} . Therefore r 1 r 3 = cos ⁡ A sin ⁡ B \dfrac{r_1}{r_3}=\dfrac{\cos {A}}{\sin {B}} . Again, A B ‾ = 2 r 2 cos ⁡ A = 2 r 1 sin ⁡ C \overline {AB}=2r_2\cos {A}=2r_1\sin {C} . So r 1 r 2 = cos ⁡ A sin ⁡ C \dfrac{r_1}{r_2}=\dfrac{\cos {A}}{\sin {C}} . Therefore r 1 : r 2 : r 3 = cos ⁡ A : sin ⁡ C : sin ⁡ B r_1:r_2:r_3=\cos {A}:\sin {C}:\sin {B}

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