Triangle Centers: Nagel Point Proof

Visual illustration
The lines connecting each vertex of a triangle ABC with the point of tangency between the opposite side and the opposite excircle are concurrent at a point called the Nagel point.



See complete interactive proof with animation and key concepts at:
www.gogeometry.com/nagel_point1.htm

Level: High School, SAT Prep, College geometry

Elearn Geometry Problem 169

See complete Problem 169 at:
www.gogeometry.com/problem/p169_parallelogram_triangle_pentagon_area.htm

Parallelogram, Interior and Exterior Points, Diagonals, Pentagon, Triangles, Areas. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 168

See complete Problem 168 at:
www.gogeometry.com/problem/p168_parallelogram_triangle_pentagon_area.htm

Parallelogram, Diagonals, Pentagon, Triangles, Areas. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 167

See complete Problem 167 at:
www.gogeometry.com/problem/p167_parallelogram_triangle_area.htm

Parallelogram, Diagonal, Triangles, Areas. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 166

See complete Problem 166 at:
www.gogeometry.com/problem/p166_parallelogram_triangle_area.htm

Parallelogram, Diagonal, Triangles, Areas. Level: High School, SAT Prep, College geometry

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Semiperimeter, Excircles, Incircle, Tangents

Visual illustration
Let ABC be a triangle of sides a, b, c and semiperimeter s. Represent (s-a), (s-b), (s-c).



See complete interactive proof with animation and key concepts at:
www.gogeometry.com/semiperimeterexcirclesinc1.htm

Level: High School, SAT Prep, College geometry

Elearn Geometry Problem 165

See complete Problem 165 at:
www.gogeometry.com/problem/p165_parallelogram_triangle_area.htm

Parallelogram, Diagonal, Triangles, Areas. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 164

See complete Problem 164 at:
www.gogeometry.com/problem/p164_parallelogram_triangle_area.htm

Parallelogram, Trapezoid, Diagonal, Triangles, Areas. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 163

See complete Problem 163 at:
www.gogeometry.com/problem/p163_trapezoid_triangle_area.htm

Trapezoid, Diagonals, Triangles, Areas. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 162

See complete Problem 162 at:
www.gogeometry.com/problem/p162_parallelogram_triangle_area.htm

Parallelogram, Triangles, Areas. Level: High School, SAT Prep, College geometry

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Golden Rectangle and Tropical Storm Fay

Visual illustration
Successive Golden Rectangles dividing a Golden Rectangle into squares (NOAA satellite image of Tropical Storm Fay over southern Florida).


See animation at:
www.gogeometry.com/wonder_world/golden_rectangle_tropical_storm_fay.html

America's Best Colleges 2009, Ranking, Interactive Mind Map

Visual illustration
America's Best Colleges 2009, Ranking, Interactive Mind Map based on : The U.S.News & World Report Magazine.


See the interactive mindmap at:
www.gogeometry.com/mindmap/america_best_colleges_2009_mind_map_elearning.html

Level: High School, SAT Prep, College geometry

Semiperimeter and excircles of a triangle

Visual illustration
Let ABC be a triangle of sides a, b, c and semiperimeter s. Represent s, (s-a), (s-b), (s-c).



See complete interactive proof with animation and key concepts at:
www.gogeometry.com/semiperimeterexcircles1.htm

Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 161

See complete Problem 161
Parallelogram, Midpoints, Octagon, Areas. Level: High School, SAT Prep, College geometry

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Semiperimeter and incircle of a triangle

Visual illustration
Let ABC be a triangle of sides a, b, c and semiperimeter s. Represent (s-a), (s-b), (s-c).



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

Post your solutions or ideas in the comments.

Elearn Geometry Problem 160

See complete Problem 160
Triangle, Incircle, Incenter, Circumcircle, Circumcenter, Inradius. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 159

See complete Problem 159
Distances from the Circumcenter to the Incenter and the Excenters. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 158

See complete Problem 158
Relation between the Circumradius, Inradius and Exradii of a triangle. Level: High School, SAT Prep, College geometry

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Pascal's Mystic Hexagram Theorem Proof

Proposition
Let A, B, C, D, E, F be points on a circle and makes a hexagon out of them in an arbitrary order. Then the three points L, M, N at which pairs of opposite sides meet, lie on a straight line. This line is called the Pascal line of the hexagon ABCDEF.



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

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Elearn Geometry Problem 157

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Distance from the Circumcenter to the Excenter. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 156

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Triangle, Circumradius, Exradius, Chord, Secant line. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 155

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Euler's Theorem: Distance from the Incenter to the Circumcenter. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 154

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Triangle, Inradius, Circumradius, Chord. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 153

See complete Problem 153
Circumscribed Quadrilateral, Diagonals Concurrent with Chords. Level: High School, SAT Prep, College geometry

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Elearn Geometry Problem 152

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Circumscribed Quadrilateral, Diagonal, Chord, Proportion. Level: High School, SAT Prep, College geometry

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Ceva's Theorem Proof

Proposition
Let ABC be a triangle and lines (or cevians) AD, BE, and CF joining vertices with opposite sides intersect at a single point P. Prove that: (AF.BD.CE)/(FB.DC.EA)=1. The converse is also true.



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

Post your solutions or ideas in the comments.