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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Solution

| Comment 1 of 2

Not having access to subscripts, I will use lowercase for the points defined in the problem Ca, Cb, Da, Db.

Angles ACB and ADB are inscribed in the same circle to the same arc and so are congruent.

By the two secants theorem, this makes arcs CaCb and DaDb congruent. And by subtracting the intersection of these arcs, CaDa and CbDb are also congruent.

The inscribed angles CbDbCa and DaCaDb are therefore congruent as are the angles CbCaDb and DaDbCa.

This means triangles DbICa and DbJCa are both isosceles with base DbCa.

So I and J both lie on the perpendicular bisector of chord DbCa, which must pass through O1, the center of the circle.

Posted by Jer on 2018-05-27 13:28:40

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