Miguel Ochoa's van Schooten Is a Slanted Viviani:
An Ultimate Generalization

What Might This Be About?

8 September 2015, Created with GeoGebra

Problem

In $\Delta ABC,$ circle $(O)$ through $A$ and cuts $BC$ in $D_1$ and $D_2$ and sides $AB$ and $AC$ in $E$ and $F,$ respectively.

Miguel Ochoa's van Schooten ultimate generalization, problem

Prove that $D_1E + D_1F = AD_2$ and $D_2E + D_2F=AD_1.$

Proof

As has been done previously, $\angle D_1FB=\angle AD_2B=\angle D_1EA.$ It thus follows from the slanted Viviani theorem that $D_1E + D_1F = AD_2.$ $D_2E + D_2F=AD_1$ is proved similarly.

Miguel Ochoa's van Schooten ultimate generalization, problem

Acknowledgment

Gobbalipur Jayanth has shown that his previous proof extends to the case where the given circle is not required to be tangent to the base $BC.$ His proof easily handles the more general case. However, as was observed by Grégoire Nicollier, both follow from the slanted Viviani theorem. Above, we have adopted Grégoire's approach.

 

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