This script constructs the Gergonne Point in a given triangle.
The Gergonne Point is the intersection point of each of the three cevians from the point of tangency of the incircle to a triangle side to its opposite apex, repectively
D - Medial Triangle.gss
An interior triangle formed from the mid-points of the sides.
D - Median Lines Triangle Construct.gss
A new triangle is constructed from 'extracting' the median lines from the given triangle.
D - Nagel Point Construction.gss
D - Nine Point Circle Centre Construction.gss
D - Nine Point Circle Construction.gss
D - Nine Point Circle Contruction and Full Labelling.gss
D - Orthic Triangle Construction (Basic).gss
D - Pedal Triangle - Circle Investigation.gss
Drop perpendiculars from the ponit D to each side (possibly extended) of the triangle. Join these intersection points to form the Pedal Triangle.
Before you run the script with your 4 points you could experiment by adding a circle and placing the point D on the circumference on this circle. Then once the Pedal Triangle has been constructed try moving the point D around its circle. In another experiment you could put the point D on the Circumcircle of the Triangle ABC before running the script.
Try constructing the Orthocentre of the Pedal Triangle, highlight it and press "Ctrl+T " - this will put a 'Trace' on this Orthocentre which can be followed as the point D is moved.
Sketches
D - Berzsenyi Triple Circle Concurrency.gsp
All three circles always meet at a single point and behind this property is a beautifully symmetric set of formulae.
D - Carnot's Theorem.gsp
Met above in Japanese Temple Geometry - A most useful Theorem.
D - Ceva's Theorem (Triangle Side Ratios).gsp
A fundamental Theorem with many hidden applications especially in proving many Triangle Centre Theorems
D - Collinearity Property of Exterior Angle Bisectors and Sides.gsp
An interesting property of extended sides and exterior angle bisector lines.
D - Collinearity Property of Orthocentres of 4 Arbitrary Lines.gsp
An amazing Theorem - and it looks particularly complicated! However, it can be simplified and explained a step at a time with Show/Hide Buttons.
D - Collinearity Property of Triangle with 3 Parallelograms.gsp
Met above in 'Quadrilaterals' page - still amazing!
D - Desargue 2D Concurrency.gsp
Desargues Theorem says that if corresponding sides of two triangles meet in three points lying on a straight line, then corresponding vertices lie on three concurrent lines.
