perplexus.info :: Geometry : Where's the Center

Where's the Center (Posted on 2018-05-20)

Circles Γ₁ and Γ₂ ( with centers O₁ and O₂ respectively ) intersect
at points A and B such that neither center lies on or inside the other circle.
Lines m_A and m_B are tangent to circle Γ₁ at points A and B respectively.

Distinct points C and D lie on Γ₂ ∩ H(A,m_B) ∩ H(B,m_A), where H(X,m_Y)
denotes the open half-plane determined by the line m_Y and does not contain point X.

Rays CA, CB, DA, and DB intersect Γ₁ at points C_A, C_B, D_A, and D_B
respectively.

   I = C_AD_A ∩ C_BD_B
                and
   J = C_AC_B ∩ D_AD_B

Prove that points I, J, and O₁ are collinear.

See The Solution Submitted by Bractals

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re: Solution

Comment 2 of 2 |

(In reply to Solution by Jer)

Nice proof.

Did you notice that you can construct the center

of the circle by adding a point E (distinct from

C and D ) only using a straightedge. Therefore,

you can perform any compass-straightedge

construction with a straightedge only if you have

two intersecting circles.

Posted by Bractals on 2018-05-27 14:22:27

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