Miguel Ochoa's van Schooten Like Theorem

What Might This Be About?

10 May 2015, Created with GeoGebra

Problem

In $\Delta ABC,$ circle $(O)$ through $A$ and tangent to $BC$ in $D$ cuts $AB$ and $AC$ in $E$ and $F,$ respectively.

Miguel Ochoa's van Schotten Like Theorem, problem

Prove that $AD = DE + DF.$

In the notations I follow the original formulation:

Miguel Ochoa's van Schotten Like Theorem, original problem

Proof 1

Rotate $A,B,C,F$ $60^{\circ}$ counterclockwise around $D$ into positions $A',B',C',F',$ respectively.

Miguel Ochoa's van Schotten Like Theorem, solution

Since $\angle EDF=120^{\circ},$ $F'$ is collinear with $D$ and $E$ so that

$\begin{align} EF'&=DE+DF'\\ &=DE+DF. \end{align}$

I am going to show that

  1. $BC'=AD,$

  2. $BC'\parallel EF'.$

Since, obviously, $A'B'\parallel AC,$ this will furnish the proof of the statement.

$CC'=CD,$ implying $BD=AC'.$ Also, $\angle BDC=\angle AC'D=120^{\circ}.$ It follows that triangles $BC'D$ and ADC' are equal, so that, in particular, $BC'=AD.$ In addition, $\angle C'BD=\angle DAC'.$ But $\angle DAC'=\angle DAC=\angle CDE,$ the former being an inscribed angle subtended by the arc $DE,$ the latter is defined by the chord $DE$ and the tangent at $D.$ Thus, $\angle C'BD=\angle EDC,$ making $BC'\parallel DE$ and, hence, $BC'\parallel EF'.$

Proof 2

Without loss of generality, set $AB=1,$ $BD=k$ and $CD=1-k,$ where $k\in(0,1).$ By the Power of a Point Theorem, $BF\cdot AB=BD^{2},$ implying $BF=k^{2}$ and $AF=1-k^2.$ By Stewart's theorem in $\Delta ABD$ and the cevian $DF,$

$FD^{2}\cdot AB+BF\cdot AF\cdot AB=BD^{2}\cdot AF+AD^{2}\cdot BF.$

In other words,

$FD^{2}\cdot AB+k^{2}(1-k^{2})=k^{2}(1-k^{2})+k^{2}AD^{2},$

i.e., $FD=kAD.$ Similarly, $DE=(1-k)AD.$ Adding up finalize the proof.

Acknowledgment

The above statement has been posted by Miguel Ochoa Sanchez at the CutTheKnotMath facebook page. The second proof is by Leo Giugiuc.

I placed proofs by Josué García Piscoya and Gobbalipur Jayanth on separate pages.

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