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

### 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. 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. ### 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.

• Angle Trisectors on Circumcircle
• Equilateral Triangles On Sides of a Parallelogram
• Pompeiu's Theorem
• Pairs of Areas in Equilateral Triangle
• The Eutrigon Theorem
• Equilateral Triangle in Equilateral Triangle
• Seven Problems in Equilateral Triangle
• Spiral Similarity Leads to Equilateral Triangle
• Parallelogram and Four Equilateral Triangles
• A Pedal Property in Equilateral Triangle
• Miguel Ochoa's van Schooten Like Theorem
• Two Conditions for a Triangle to Be Equilateral
• Incircle in Equilateral Triangle
• When Is Triangle Equilateral: Marian Dinca's Criterion
• Barycenter of Cevian Triangle
• Excircle in Equilateral Triangle
• Converse Construction in Pompeiu's Theorem
• Wonderful Trigonometry In Equilateral Triangle
• 60o Angle And Importance of Being The Other End of a Diameter
• One More Property of Equilateral Triangles
• Van Khea's Quickie
• Equilateral Triangle from Three Centroids