Thursday, July 31, 2008

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.



See complete interactive proof with animation and key concepts
Level: High School, SAT Prep, College geometry

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Sunday, July 27, 2008

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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Friday, July 25, 2008

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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Thursday, July 24, 2008

Elearn Geometry Problem 142

Four triangles, incircle, areas

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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

 Triangle, excircle, semiperimeter

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

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Tuesday, July 22, 2008

Elearn Geometry Problem 139

Area of triangle

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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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Monday, July 21, 2008

Elearn Geometry Problem 137

Geometry Problem 137 Concyclic points

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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

Geometry Problem 136 Orthic triangle

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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

Go Geometry Problem: Orthic triangle

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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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Friday, July 11, 2008

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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Thursday, July 10, 2008

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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Wednesday, July 9, 2008

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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Tuesday, July 8, 2008

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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Monday, July 7, 2008

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.
Prove proposition.. Level: High School, SAT Prep, College geometry

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Thursday, July 3, 2008

Dynamic Geometry: Triangle

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

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Wednesday, July 2, 2008

Elearn Geometry Problem 133



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

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