Menelaus' Theorem Proof

Proposition
Let ABC be a triangle and MEN a line that transverse (crosses) the lines AB, BC, and AC respectively. Prove that (AM.BE.CN)/(MB.EC.NA)=-1. The converse also holds.



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Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 151

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Quadrilateral, Area, Trisection of Sides. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 150

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Quadrilateral, Area, Trisection of Sides. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 149

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Quadrilateral, Area, Midpoints. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 148

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Quadrilateral, Area, Midpoints. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 147

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Quadrilateral, Area, Midpoints, Triangle. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 146 Varignon s theorem and more

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Varignon's Theorem: Quadrilateral, Midpoints, Parallelogram, Area, Perimeter. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 145

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Four Triangles, Incircle, Tangent and Parallel to Side, Incenters, Circumcenters. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 144

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Four Triangles, Incircle, Tangent and Parallel to Side, Circumradii. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 143

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Four Triangles, Incircle, Tangent and Parallel to Side, Circumradii. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 142

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Four Triangles, Incircle, Tangent and Parallel to Side, Areas. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 141

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Triangle, Incircle, Tangent and parallel to side, Perimeter. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 140

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Triangle, Excircle, Tangent, Semiperimeter. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 139

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Triangle Area, Orthic Triangle, Semiperimeter, Circumradius. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 138 Nagel's Theorem

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Nagel's Theorem, Orthic Triangle, Altitudes, Circumradius, Perpendicular. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 137

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Orthic Triangle, Altitudes, Perpendicular, Incircle, Collinear Points, Parallelogram. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 136

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Orthic Triangle, Altitudes, Orthocenter, Incenter, Perpendicular, Concyclic Points. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 135

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Orthic Triangle, Altitudes, Perpendicular, Parallel. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 134

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Orthic Triangle, Altitudes, Angle Bisectors, Orthocenter, Incenter. Level: High School, SAT Prep, College geometry

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Pappus Theorem

Proposition
If six points 1,2,3,4,5, and 6, on two lines are joined as shown, then their points of intersection A, B, and C are collinear.



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Level: High School, SAT Prep, College geometry

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Gergonne Line

Proposition
Given a triangle ABC, D, E, and F are the points of contact of the incircle with the sides, as shown. AC and DF meets at B', BC and EF meets at A', and AB and DE meets at C'. Prove that A', B', and C' are collinear.

The line A'B'C' is called Gergonne Line and the points A',B'C' are called Nobbs' points.



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Level: High School, SAT Prep, College geometry

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Interactive Simson Line

Proposition
Given a triangle ABC and P a point on its circumcircle, as shown. Prove that the feet D, E, and F of the perpendiculars drawn from P to the sides (or their extensions) are collinear. The line DEF is called the Simson line.



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Triangle, Circumcircle, Perpendiculars, Collinear feet points. Level: High School, SAT Prep, College geometry

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Triangle, Medians, Six Circumcenters Concyclic

Proposition
Medians AD, BE, and CF split a triangle ABC into six smaller triangles. Prove that their circumcenters O1, O2, O3, O4, O5, and O6 are concyclic (lie on a circle).



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Triangle, Medians, Six Circumcenters Concyclic. Level: High School, SAT Prep, College geometry

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Trapezoid, diagonal, midpoints

Proposition
Given a triangle ABC and a trapezoid ADEC (DE // AC), as shown. AE and CD meet at F, and BF meets DE and AC at N and M respectively. Prove that M and N are the midpoints of AC and DE respectively.



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Trapezoid, Triangle, Diagonals, Midpoints. Dynamic Geometry.
Step-by-Step construction, Manipulation, and animation.
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Dynamic Geometry: Triangle

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Triangle: Incircle, Perpendicular, Angle Bisector. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 133

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Triangle, Angle Bisectors, Collinear Points. Level: High School, SAT Prep, College geometry

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